Compilation of Mathematicians and Their Contributions

I. Greek Mathematicians Thales of Miletus Birthdate 624 B. C. Died 547-546 B. C. Nationality Greek Title Regarded as Father of Science Contributions * He is credited with the commencement exercise use of deductive reasoning apply to geometry. * Discovery that a circle isbisectedby its diameter, that the base angles of an isosceles trigon atomic subject 18 equal and that upright piano angles be equal. * Accredited with keister of the Ionian shoal of Mathematics that was a centre of learning and research. * Thales theorems used in Geometry . The pairs of opposite angles formed by devil intersecting lines are equal. 2. The base angles of an isosceles triplicity are equal. 3. The sum of the angles in a triangle is 180. 4. An angle inscribed in a semicircle is a right angle. Pythagoras Birthdate 569 B. C. Died 475 B. C. Nationality Greek Contributions * Pythagorean Theorem. In a right angled triangle the square of the hypo xuse is equal to the sum of the squares on the otherwise two si stilboestrol. Note A right triangle is a triangle that contains oneness right (90) angle.The longest side of a right triangle, call(a)ed the hypotenuse, is the side opposite the right angle. The Pythagorean Theorem is classical in math, physics, and astronomy and has practical applications in surveying. * Developed a sophisticated numerology in which odd payoffs de noned mannish and even female 1 is the generator of add up and is the number of reason 2 is the number of sight 3 is the number of harmony 4 is the number of justice and retribution (opinion squared) 5 is the number of marriage (union of the ? rst male and the ? st female numbers) 6 is the number of creation 10 is the holiest of all, and was the number of the universe, because 1+2+3+4 = 10. * Discovery of incommensurate ratios, what we would call instantly anomalous numbers. * Made the ? rst inroads into the branch of mathematics which would today be called Number Theory. * Setting up a secret mystical so ciety, k instanter as the Pythagoreans that taught Mathematics and Physics. Anaxagoras Birthdate vitamin D B. C. Died 428 B. C. Nationality Greek Contributions * He was the outgrowth to explain that the moon shines due to reflected light from the sun. Theory of minute constituents of things and his emphasis on mechanical processes in the formation of parliamentary procedure that paved the instruction for the atomic hypothesis. * Advocated that matter is composed of in delimited elements. * Introduced the notion of nous (Greek, forefront or reason) into the philosophy of origins. The apprehension of nous (mind), an infinite and unchanging substance that enters into and controls all living object. He regarded material substance as an infinite multitude of imperish subject primary elements, referring all extension and disappearance to mixture and separation, respectively.Euclid Birthdate c. 335 B. C. E. Died c. 270 B. C. E. Nationality Greek Title Father of Geometry Contributi ons * create a track record called the Elements serving as the main text disk for t all(prenominal)ingmathematics(especiallygeometry) from the period of its publication until the late 19th or early 20th century. The Elements. One of the oldest surviving fragments of EuclidsElements, found atOxyrhynchus and dated to circa AD 100. * Wrote take shapes on perspective, conic section section sections,spherical geometry,number possibilityandrigor. In addition to theElements, at least five works of Euclid take over survived to the present day. They follow the same logical social organisation asElements, with definitions and turn out propositions. Those are the following 1. Datadeals with the nature and implications of addicted information in geometrical problems the subject matter is closely related to the number one four books of theElements. 2. On Divisions of Figures, which survives and partially inArabictranslation, concerns the division of geometrical figures into two or more than equal move or into parts in givenratios.It is similar to a third century AD work byHeron of Alexandria. 3. Catoptrics, which concerns the mathematical guess of mirrors, detailly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor and E F Robertson who nameTheon of Alexandriaas a more likely author. 4. Phaenomena, a treatise onspherical astronomy, survives in Greek it is quite similar toOn the Moving SpherebyAutolycus of Pitane, who flourished around 310 BC. * historied five postulates of Euclid as mentioned in his book Elements . Point is that which has no part. 2. Line is a breadthless length. 3. The extremities of lines are manoeuvers. 4. A straight line lies equally with respect to the menstruations on itself. 5. One can draw a straight line from any point to any point. * TheElements likewise include the following five common notions 1. Things that are equal to the same thing are in any case equal to one another (Transitive property of equality). 2. If equals are added to equals, then the wholes are equal. 3. If equals are subtracted from equals, then the go forwardders are equal. 4.Things that coincide with one another equal one another (Reflexive Property). 5. The whole is heavy(p)er than the part. Plato Birthdate 424/423 B. C. Died 348/347 B. C. Nationality Greek Contributions * He helped to distinguish between arrant(a)andapplied mathematicsby widening the gap between arithmetic, at once callednumber theoryand logistic, now calledarithmetic. * Founder of theAcademyinAthens, the beginning inception of higher learning in theWestern world. It provided a comprehensive curriculum, including such subjects as astronomy, biology, mathematics, political theory, and philosophy. Helped to lay the foundations ofWestern philosophyandscience. * Platonic squares Platonic significant is a regular, convex polyhedron. The faces are congruent, regular polygons, with the same num ber of faces meeting at distributively vertex. There are exactly five solids which meet those criteria each is named according to its number of faces. * Polyhedron Vertices Edges FacesVertex configuration 1. tetrahedron4643. 3. 3 2. cube / hexahedron81264. 4. 4 3. octahedron61283. 3. 3. 3 4. dodecahedron2030125. 5. 5 5. icosahedron1230203. 3. 3. 3. 3 AristotleBirthdate 384 B. C. Died 322 BC (aged 61 or 62) Nationality Greek Contributions * Founded the Lyceum * His biggest parcel to the theater of mathematics was his nurture of the study of logic, which he termed analytics, as the basis for mathematical study. He wrote extensively on this concept in his work anterior Analytics, which was produce from Lyceum lecture notes several hundreds of years after his death. * Aristotles Physics, which contains a discussion of the infinite that he believed existed in theory only, sparked oftentimes debate in later centuries.It is believed that Aristotle may have been the number one phil osopher to draw the distinction between actual and potential infinity. When considering both actual and potential infinity, Aristotle states this 1. A body is outlined as that which is bounded by a surface, on that pointfore there cannot be an infinite body. 2. A Number, Numbers, by definition, is countable, so there is no number called infinity. 3. Perceptible bodies exist aboutwhere, they have a place, so there cannot be an infinite body. alone Aristotle says that we cannot say that the infinite does not exist for these reasons 1.If no infinite, magnitudes go out not be divisible into magnitudes, but magnitudes can be divisible into magnitudes (potentially infinitely), therefore an infinite in some sense exists. 2. If no infinite, number would not be infinite, but number is infinite (potentially), therefore infinity does exist in some sense. * He was the founder offormal logic, pioneered the study ofzoology, and left every future scientist and philosopher in his debt through his contributions to the scientific order. Erasthosthenes Birthdate 276 B. C. Died 194 B. C. Nationality Greek Contributions * Sieve of Eratosthenes Worked on outpouring numbers.He is remembered for his prime number sieve, the Sieve of Eratosthenes which, in modified form, is still an distinguished tool innumber theoryresearch. Sieve of Eratosthenes- It does so by iteratively accentuateing as composite (i. e. not prime) the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same passing, equal to that prime, between consecutive numbers. This is the Sieves key distinction from employ mental test division to sequentially test each candidate number for divisibility by each prime. Made a surprisingly accurate measurement of the circumference of the terra firma * He was the initiatory-year individual to use the word geography in Greek and he invented the discipline o f geography as we understand it. * He invented a system oflatitudeandlongitude. * He was the first to conduct thetilt of the Earths axis( as hygienic as with remarkable accuracy). * He may also have accurately calculated the keep from the earth to the sunand invented theleap day. * He also created the firstmap of the worldincorporating parallels and meridians within his cartographic depictions based on the available geographical cognition of the era. Founder of scientificchronology. Favourite Mathematician Euclid paves the way for what we known today as Euclidian Geometry that is considered as an indispensable for everyone and should be studied not only by students but by everyone because of its vast applications and relevance to everyones daily life. It is Euclid who is gifted with knowledge and therefore became the pillar of todays success in the house of geometry and mathematics as a whole. There were great mathematicians as there were numerous great mathematical knowledge tha t God wants us to know.In consideration however, there were several sagacious Greek mathematicians that had imparted their great contributions and therefore they deserve to be appreciated. But since my task is to declare my ducky mathematician, Euclid deserves more or less of my kudos for laying scratch off the foundation of geometry. II. Mathematicians in the Medieval Ages Leonardo of Pisa Birthdate 1170 Died 1250 Nationality Italian Contributions * Best known to the raw world for the spreading of the HinduArabic numeral system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation). Fibonacci introduces the supposed Modus Indorum (method of the Indians), today known as Arabic numerals. The book advocated numeration with the digits 09 and place value. The book showed the practical importance of the new numeral system, using lattice multiplication and Egyptian fractions, by applying it to commercial bookkeeping, conversion of weights and me asures, the weighing of interest, money-changing, and other applications. * He introduced us to the bar we use in fractions, previous to this, the numerator has quotations around it. * The square root greenback is also a Fibonacci method. He wrote following books that deals Mathematics teachings 1. Liber Abbaci (The Book of Calculation), 1202 (1228) 2. Practica Geometriae (The recital of Geometry), 1220 3. Liber Quadratorum (The Book of Square Numbers), 1225 * Fibonacci sequence of numbers in which each number is the sum of the previous two numbers, starting with 0 and 1. This sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 The higher up in the sequence, the closer two consecutive Fibonacci numbers of the sequence divided by each other will approach the golden ratio (approximately 1 1. 18 or 0. 618 1). Roger Bacon Birthdate 1214 Died 1294 Nationality English Contributions * Opus Majus contains treatments of mathematics and optics, alchemy, and the positions and sizes of the celestial bodies. * Advocated the experimental method as the true foundation of scientific knowledge and who also did some work in astronomy, chemistry, optics, and machine design. Nicole Oresme Birthdate 1323 Died July 11, 1382 Nationality french Contributions * He also develop a language of ratios, to relate speed to force and resistance, and applied it to physical and cosmological questions. He made a careful study of musicology and used his findings to develop the use of monstrous exponents. * First to theorise that sound and light are a transfer of energy that does not displace matter. * His almost all important(predicate) contributions to mathematics are contained in Tractatus de configuratione qualitatum et motuum. * Developed the first use of powers with aliquot exponents, calculation with irrational proportions. * He proved the divergence of the harmonic serial, using the standard method still taught in chalkstone classes today. Omar Khay yam Birhtdate 18 May 1048Died 4 December 1131 Nationality Arabian Contibutions * He derived resolvings to cubic equations using the intersection of conic sections with circles. * He is the author of one of the most important treatises on algebra indite before modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle. * He contributed to a calendar reform. * Created important works on geometry, specifically on the theory of proportions. Omar Khayyams geometric solution to cubic equations. Binomial theorem and extraction of grow. * He may have been first to develop Pascals Triangle, along with the essential Binomial Theorem which is sometimes called Al-Khayyams Formula (x+y)n = n ? xkyn-k / k (n-k). * Wrote a book entitled Explanations of the gruellingies in the postulates in Euclids Elements The treatise of Khayyam can be considered as the first treatment of parallels axiom which is not based on petitio principii but on more intuitive postulate. Khayyam refutes the previous attempts by other Greek and Iranian mathematicians to prove the proposition.In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate. Favorite Mathematician As far as medieval times is concerned, people in this era were challenged with chaos, social turmoil, economic issues, and many other disputes. Part of this era is tinted with so called phantasm Ages that marked the history with unfavourable events. Therefore, mathematicians during this era-after they undergone the untold toils-were deserving individuals for gratitude and praises for they had supplemented the following generations with mathematical ideas that is very expedient and applicable.Leonardo Pisano or Leonardo Fibonacci caught my attention therefore he is my preferent mathematician in the medieval times. His desire to spread out the Hindu-Arabic numerals in other countries thus signifies that he is a person of generosity, with his noble will, he deserves to be III. Mathematicians in the Renaissance Period Johann Muller Regiomontanus Birthdate 6 June 1436 Died 6 July 1476 Nationality German Contributions * He completed De Triangulis omnimodus. De Triangulis (On Triangles) was one of the first schoolbooks presenting the current state of trigonometry. His work on arithmetic and algebra, Algorithmus Demonstratus, was among the first containing symbolic algebra. * De triangulis is in five books, the first of which gives the basic definitions quantity, ratio, equality, circles, arcs, chords, and the sine subroutine. * The crater Regiomontanus on the Moon is named after him. Scipione del Ferro Birthdate 6 February 1465 Died 5 November 1526 Nationality Italian Contributions * Was the first to process the cubic equation. * Contributions to the rationalization of fractions with denominators containing sums of cube roots. Investigat ed geometry problems with a compass set at a fixed angle. Niccolo Fontana Tartaglia Birthdate 1499/1 calciferol Died 13 December 1557 Nationality Italian Contributions He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs his work was later validated by Galileos studies on fall bodies. He also published a treatise on retrieving sunken ships. Cardano-Tartaglia Formula. He makes solutions to cubic equations. Formula for solving all types of cubic equations, involving first real use of abstruse numbers (combinations of real and imaginary numbers). Tartaglias Triangle (earlier version of Pascals Triangle) A triangular pattern of numbers in which each number is equal to the sum of the two numbers immediately above it. He gives an expression for the volume of a tetrahedron Girolamo Cardano Birthdate 24 family li ne 1501 Died 21 phratry 1576 Nationality Italian Contributions * He wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music. Was the first mathematician to make systematic use of numbers less than zero. * He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna. * Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem. * His book about games of chance, Liber de ludo aleae (Book on Games of Chance), written in 1526, but not published until 1663, contains the first systematic treatment of probability. * He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic ircles and were used for the kink of the first high-speed printing presses. * His book, Liber de ludo aleae (Book on Games of Chance), contains the first systematic treatment of probability. * Cardanos Ring Puzzle also known as Chinese Rings, still manufactured today and related to the Tower of Hanoi puzzle. * He introduced binomial coefficients and the binomial theorem, and introduced and solved the geometric hypocyloid problem, as well as other geometric theorems (e. g. the theorem underlying the 21 spur wheel which converts circular to reciprocal rectilinear motion).Binomial theorem-formula for multiplying two-part expression a mathematical formula used to calculate the value of a two-part mathematical expression that is squared, cubed, or raised to another power or exponent, e. g. (x+y)n, without explicitly multiplying the parts themselves. Lodovico Ferrari Birthdate February 2, 1522 Died October 5, 1565 Nationality Italian Contributions * Was mainly responsible for the solution of quartic equations. * Ferrari aided Cardano on his solutions for quadratic equation equations and cubic equations, and was mainly responsible for the solution of quartic equations that Cardano published.As a result, mathem aticians for the next several centuries tried to find a formula for the roots of equations of degree five and higher. Favorite Mathematician Indeed, this period is supplemented with great mathematician as it moved on from the Dark Ages and undergone a rebirth. Enumerated mathematician were all astounding with their performances and contributions. But for me, Niccolo Fontana Tartaglia is my favourite mathematician not only because of his undisputed contributions but on the way he keep himself calm despite of conflicts between him and other mathematicians in this period. IV. Mathematicians in the 16th CenturyFrancois Viete Birthdate 1540 Died 23 February 1603 Nationality French Contributions * He true the first infinite- intersection point formula for ?. * Vieta is most famous for his systematic use of decimal notation and variable letters, for which he is sometimes called the Father of Modern Algebra. (Used A,E,I,O,U for unknowns and consonants for parameters. ) * Worked on geometry and trigonometry, and in number theory. * Introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin (nq) and cos (nq) in terms of the powers of sin(q) and cos(q). * make Francisci Viet? universalium inspectionum ad canonem mathematicum liber singularis a book of trigonometry, in abbreviated Canonen mathematicum, where there are many formulas on the sine and cosine. It is unusual in using decimal numbers. * In 1600, numbers potestatum ad exegesim proclamationer, a work that provided the means for extracting roots and solutions of equations of degree at most 6. John Napier Birthdate 1550 Birthplace Merchiston Tower, Edinburgh conclusion 4 April 1617 Contributions * Responsible for pass on the notion of the decimal fraction by introducing the use of the decimal point. His suggestion that a simple point could be used to eparate whole number and fractional parts of a number soon became accepted practice throughout Great Britain. * Inve ntion of the Napiers Bone, a crude hand calculator which could be used for division and root extraction, as well as multiplication. * Written Works 1. A Plain Discovery of the Whole Revelation of St. John. (1593) 2. A definition of the Wonderful Canon of Logarithms. (1614) Johannes Kepler Born December 27, 1571 Died November 15, 1630 (aged 58) Nationality German Title Founder of Modern Optics Contributions * He generalized Alhazens Billiard Problem, developing the notion of curvature. He was first to notice that the set of Platonic regular solids was incomplete if concave solids are admitted, and first to prove that there were only 13 Archimedean solids. * He proved theorems of solid geometry later discovered on the famous palimpsest of Archimedes. * He rediscovered the Fibonacci series, applied it to botany, and noted that the ratio of Fibonacci numbers converges to the Golden Mean. * He was a key early pioneer in calculus, and embraced the concept of continuity (which others avo ided due to Zenos paradoxes) his work was a direct inspiration for Cavalieri and others. He developed mensuration methods and anticipated Fermats theorem (df(x)/dx = 0 at function extrema). * Keplers Wine Barrel Problem, he used his rudimentary calculus to deduce which barrel shape would be the best bargain. * Keplers Conjecture- is a mathematical hypothesize about sphere packing in 3-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater median(a) density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.Marin Mersenne Birthdate 8 September 1588 Died 1 September 1648 Nationality French Contributions * Mersenne primes. * Introduced several innovating concepts that can be considered as the basis of modern reflecting telescopes 1. Instead of using an eyepiece, Mersenne introduced the revolutionary idea of a stake mirror that would reflect the light coming from the first mirror. This allows one to focus the image behind the primary mirror in which a hole is drilled at the centre to unblock the rays. 2.Mersenne invented the afocal telescope and the beam compressor that is helpful in many multiple-mirrors telescope designs. 3. Mersenne accept also that he could correct the spherical aberration of the telescope by using nonspherical mirrors and that in the particular case of the afocal arrangement he could do this correction by using two parabolic mirrors. * He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 1644.He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulums swings are not isochronous as Galileo thought, but that large swings take longer than small swings. Gerard Desargues Birthdate February 21, 1591 Died September 1661 Nationality French Contributions * Founder of the theory of conic sections. Desargues offered a unified approach to the several types of conics through projection and section. * Perspective Theorem that when two triangles are in perspective the meets of corresponding sides are collinear. * Founder of projective geometry. Desarguess theorem The theorem states that if two triangles ABC and A? B? C? , situated in trine-dimensional space, are related to each other in such a way that they can be seen perspectively from one point (i. e. , the lines AA? , BB? , and CC? all intersect in one point), then the points of intersection of corresponding sides all lie on one line provided that no two corresponding sides are * Desargues introduced the notions of the opposite ends of a straight line being regarded as coincident, parallel lines meeting at a point of infinity and regarding a straight line as circle whose center is at infinity. Desargues most important work Brouillon projet dune atte inte aux evenemens des rencontres d? une cone avec un plan (Proposed Draft for an essay on the results of taking plane sections of a cone) was printed in 1639. In it Desargues presented innovations in projective geometry applied to the theory of conic sections. Favorite Mathematician Mathematicians in this period has its own distinct, and unique knowledge in the field of mathematics.They tackled the more complex world of mathematics, this complex world of Mathematics had at times stirred their lives, ignited some conflicts between them, unfolded their flaws and weaknesses but at the end, they build harmonious world through the unity of their formulas and much has benefited from it, they indeed reflected the bang of Mathematics. They were all excellent mathematicians, and no doubt in it. But I admire John Napier for giving birth to Logarithms in the world of Mathematics. V. Mathematicians in the seventeenth Century Rene Descartes Birthdate 31 March 1596 Died 11 February 1650National ity French Contributions * Accredited with the invention of co-ordinate geometry, the standard x,y co-ordinate system as the Cartesian plane. He developed the coordinate system as a device to locate points on a plane. The coordinate system includes two perpendicular lines. These lines are called axes. The vertical axis is designated as y axis while the horizontal axis is designated as the x axis. The intersection point of the two axes is called the origin or point zero. The position of any point on the plane can be located by locating how far perpendicularly from each axis the point lays.The position of the point in the coordinate system is specified by its two coordinates x and y. This is written as (x,y). * He is credited as the give of analytical geometry, the bridge between algebra and geometry, crucial to the baring of infinitesimal calculus and psychoanalysis. * Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of ge nius. * He also pioneered the standard notation that uses superscripts to show the powers or exponents for example, the 4 used in x4 to indicate squaring of squaring. He invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c. * He was first to assign a fundamental place for algebra in our system of knowledge, and believed that algebra was a method to automate or motorize reasoning, particularly about abstract, unknown quantities. * Rene Descartes created analytic geometry, and discovered an early form of the law of conservation of momentum (the term momentum refers to the momentum of a force). * He developed a rule for determining the number of positive and negative roots in an equation.The Rule of Descartes as it is known states An equation can have as many true positive roots as it contains salmagundis of sign, from + to or from to + and as many false negative roots as the number of times two + signs or two signs are found in su ccession. Bonaventura Francesco Cavalieri Birthdate 1598 Died November 30, 1647 Nationality Italian Contributions * He is known for his work on the problems of optics and motion. * Work on the precursors of infinitesimal calculus. * Introduction of logarithms to Italy. First book was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections. In this book he developed the theory of mirrors shaped into parabolas, hyperbolas, and ellipses, and various combinations of these mirrors. * Cavalieri developed a geometrical approach to calculus and published a treatise on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635).In this work, an area is considered as comprise by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. * Cavalieris principle, which state s that the volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the body with planes equidistant from a chosen base plane. * Published tables of logarithms, punctuate their practical use in the handle of astronomy and geography.Pierre de Fermat Birthdate 1601 or 1607/8 Died 1665 Jan 12 Nationality French Contributions * Early developments that led to infinitesimal calculus, including his proficiency of adequality. * He is recognized for his discovery of an passe-partout method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. * He made notable contributions to analytic geometry, probability, and optics. * He is best known for Fermats Last Theorem. Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. * He invented a factorization methodFermats factorization methodas well as the proof technique of infinite descent, which he used to prove Fermats Last Theorem for the case n = 4. * Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of terce triangular numbers, four square numbers, five pentagonal numbers, and so on. With his gift for number sexual intercourses and his ability to find proofs for many of his theorems, Fermat fundamentally created the modern theory of numbers. Blaise Pascal Birthdate 19 June 1623 Died 19 August 1662 Nationality French Contributions * Pascals Wager * Famous contribution of Pascal was his Traite du triangle arithmetique (Treatise on the Arithmetical Triangle), commonly known today as Pascals triangle, which demonstrates many mathematical properties like binomial coefficients. Pascals Triangle At the age of 16, he formulated a basic theorem of projective geometry, known today as Pascals theorem. * Pascals law (a hydrostatics principle). * He invented the mechanical calculator. He built 20 of these machines (called Pascals calculator and later Pascaline) in the following ten years. * Corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. * Pascals theorem. It states that if a hexagon is inscribed in a circle (or conic) then the three intersection points of opposite sides lie on a line (called the Pascal line).Christiaan Huygens Birthdate April 14, 1629 Died July 8, 1695 Nationality Dutch Contributions * His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan. * The invention of the pendulum clock. Spring driven pendulum clock, designed by Huygens. * Discovery of the centrifugal force, the laws for collision of bodies, for his role in the development of modern calculus and his original observations on sound perception. Wrote the first book on probability theory, De ratiociniis in ludo aleae (On Reasoning in Games of Chance). * He also designed more accurate clocks than were available at the time, suitable for sea sailplaning. * In 1673 he published his mathematical analysis of pendulums, Horologium Oscillatorium sive de motu pendulorum, his greatest work on horology. Isaac due north Birthdate 4 Jan 1643 Died 31 March 1727 Nationality English Contributions * He laid the foundations for differential and integral calculus.Calculus-branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics ca lled analysis. * Produced simple analytical methods that unified many separate techniques previously developed to solve evidently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions. Investigated the theory of light, explained gravity and hence the motion of the planets. * He is also famed for inventing Newtonian Mechanics and explicating his famous three laws of motion. * The first to use fractional indices and to wage coordinate geometry to derive solutions to Diophantine equations * He discovered Newtons identities, Newtons method, classified cubic plane curves (multinomials of degree three in two variables) Newtons identities, also known as the NewtonGirard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials.Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots * Newtons method (also known as the NewtonRaphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. Gottfried Wilhelm Von Leibniz Birthdate July 1, 1646 Died November 14, 1716 Nationality GermanContributions * Leibniz invented a mechanical astute machine which would multiply as well as add, the mechanics of which were still being used as late as 1940. * Developed the infinitesimal calculus. * He became one of the most prolific inventors in the field of mechanical calculators. * He was the first to describe a pinwheel calculator in 16856 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. * He also refined the binary number system, which is at the foundation of virtually all digital computers. Leibniz was the firs t, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular. * Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system. * He introduced several notations used to this day, for instance the integral sign ? representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia.This cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. * He was the ? rst to use the notation f(x). * The notation used today in Calculus df/dx and ? f x dx are Leibniz notation. * He also did work in discrete mathematics and the foundations of logic. Favorite Mathematician Selecting favourite mathematician from these adept persons in mathematics is a hard task, but as I read the contributions of these Mathematicians, I found Sir Isaac Newton to be the greatest mathematician of this period.He invented the useful but difficult subject in mathematics- the calculus. I found him cooperative with different mathematician to derive useful formulas despite the fact that he is bright enough. Open-mindedness towards others opinion is what I discerned in him. VI. Mathematicians in the 18th Century Jacob Bernoulli Birthdate 6 January 1655 Died 16 August 1705 Nationality Swiss Contributions * Founded a school for mathematics and the sciences. * Best known for the work Ars Conjectandi (The Art of Conjecture), published eight years after his death in 1713 by his nephew Nicholas. Jacob Bernoullis first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. * Introduction of the theorem known as the law of large numbers. * By 1689 he had published important work on i nfinite series and published his law of large numbers in probability theory. * Published five treatises on infinite series between 1682 and 1704. * Bernoulli equation, y = p(x)y + q(x)yn. * Jacob Bernoullis paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. Discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. * Theory of permutations and combinations the so-called Bernoulli numbers, by which he derived the exponential series. * He was the first to think about the convergence of an infinite series and proved that the series is convergent. * He was also the first to propose continuously compounded interest, which led him to investigate Johan Bernoulli Birthdate 27 July 1667Died 1 January 1748 Nationality Swiss Contributions * He was a brilliant mathematician who made important discoveries in the field of calculus. * He is known for his contributions to infinitesimal calculus and educated Leonhard Euler in his youth. * Discovered fundamental principles of mechanics, and the laws of optics. * He discovered the Bernoulli series and made advances in theory of navigation and ship sailing. * Johann Bernoulli proposed the brachistochrone problem, which asks what shape a wire must be for a bead to slide from one end to the other in the shortest possible time, as a challenge to other mathematicians in June 1696.For this, he is regarded as one of the founders of the calculus of variations. Daniel Bernoulli Birthdate 8 February 1700 Died 17 March 1782 Nationality Swiss Contributions * He is particularly remembered for his applications of mathematics to mechanics. * His pioneering work in probability and statistics. Nicolaus Bernoulli Birthdate February 6, 1695 Died July 31, 1726 Na tionality Swiss Contributions Worked mostly on curves, differential equations, and probability. He also contributed to fluid dynamics. Abraham de Moivre Birthdate 26 May 1667 Died 27 November 1754 Nationality French Contributions Produced the second textbook on probability theory, The Doctrine of Chances a method of calculating the probabilities of events in play. * Pioneered the development of analytic geometry and the theory of probability. * Gives the first statement of the formula for the general distribution curve, the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the division of the distribution as a unit, and the first identification of the probable error calculation. Additionally, he applied these theories to gambling problems and actuarial tables. In 1733 he proposed the formula for estimating a factorial as n = cnn+1/2e? n. * Published an article called Annuities upon Lives, in which he reve aled the normal distribution of the mortality rate over a persons age. * De Moivres formula which he was able to prove for all positive integral values of n. * In 1722 he suggested it in the more well-known form of de Moivres Formula Colin Maclaurin Birthdate February, 1698 Died 14 June 1746 Nationality Scottish Contributions * Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. Made significant contributions to the gravitation magnet of ellipsoids. * Maclaurin discovered the EulerMaclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirlings formula, and to derive the Newton-Cotes numerical integration formulas which includes Simpsons rule as a special case. * Maclaurin contributed to the study of oviform integrals, reducing many intractable integrals to problems of finding arcs for hyperbolas. * Maclaurin proved a rule for solving square linear system s in the cases of 2 and 3 unknowns, and discussed the case of 4 unknowns. Some of his important works are Geometria Organica 1720 * De Linearum Geometricarum Proprietatibus 1720 * Treatise on Fluxions 1742 (763 pages in two volumes. The first systematic exposition of Newtons methods. ) * Treatise on Algebra 1748 (two years after his death. ) * Account of Newtons Discoveries uncomplete upon his death and published in 1750 or 1748 (sources disagree) * Colin Maclaurin was the name used for the new Mathematics and Actuarial Mathematics and Statistics Building at Heriot-Watt University, Edinburgh. Lenard Euler Birthdate 15 April 1707 Died 18 September 1783 Nationality Swiss Contributions He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. * He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. * He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. * Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function 2 and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Eulers number), the Greek letter ? for summations and the letter i to denote the imaginary unit. * The use of the Greek letter ? to denote the ratio of a circles circumference to its diameter was also popularized by Euler. * Well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as * Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully jelld logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. * He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. * Elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis.He also invented the calculus of variations including its best-known result, the EulerLagrange equation. * Pioneered the use of analytic methods to solve number theory problems. * Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Eulers work in this area led to the development of the prime number theorem. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers this is known as the Euler product formula for the Riemann zeta function. * He also invented the totient function ? (n) which is the number of positive integers less than or equal to the integer n that are coprime to n. * Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. * Discovered the formula V ?E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron. * He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. Jean Le Rond De Alembert Birthdat e 16 November 1717 Died 29 October 1783 Nationality French Contributions * DAlemberts formula for obtaining solutions to the wave equation is named after him. * In 1743 he published his most famous work, Traite de dynamique, in which he developed his own laws of motion. * He created his ratio test, a test to see if a series converges. The DAlembert operator, which first arose in DAlemberts analysis of vibrating strings, plays an important role in modern theoretical physics. * He made several contributions to mathematics, including a suggestion for a theory of limits. * He was one of the first to appreciate the importance of functions, and defined the derived of a function as the limit of a quotient of increments. Joseph Louise Lagrange Birthdate 25 January 1736 Died 10 April 1813 Nationality Italian French Contributions * Published the Mecanique Analytique which is considered to be his monumental work in the pure maths. His most prominent influence was his contribution to the the m etric system and his addition of a decimal base. * Some refer to Lagrange as the founder of the Metric System. * He was responsible for developing the groundwork for an alternate method of writing Newtons Equations of Motion. This is referred to as Lagrangian Mechanics. * In 1772, he described the Langrangian points, the points in the plane of two objects in field around their common center of gravity at which the combined gravitational forces are zero, and where a third particle of negligible mass can remain at rest. He made significant contributions to all fields of analysis, number theory, and classical and celestial mechanics. * Was one of the creators of the calculus of variations, deriving the EulerLagrange equations for extrema of functionals. * He also lengthened the method to take into account possible constraints, arriving at the method of Lagrange multipliers. * Lagrange invented the method of solving differential equations known as variation of parameters, applied diffe rential calculus to the theory of probabilities and attained notable work on the solution of equations. * He proved that every natural number is a sum of four squares. some(prenominal) of his early papers also deal with questions of number theory. 1. Lagrange (17661769) was the first to prove that Pells equation has a nontrivial solution in the integers for any non-square natural number n. 7 2. He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770. 3. He proved Wilsons theorem that n is a prime if and only if (n ? 1) + 1 is always a multiple of n, 1771. 4. His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved. 5.His Recherches dArithmetique of 1775 developed a general theory of binary quadratic forms to clench the general problem of when an integer is representable by the form. Gaspard Monge Birthdate May 9, 1746 Died July 28, 1818 Nationality F rench Contributions * Inventor of descriptive geometry, the mathematical basis on which practiced drawing is based. * Published the following books in mathematics 1. The Art of Manufacturing Cannon (1793)3 2. Geometrie descriptive. Lecons donnees aux ecoles normales (Descriptive Geometry) a transcription of Monges lectures. (1799) Pierre Simon Laplace Birthdate 23 March 1749Died 5 March 1827 Nationality French Contributions * Formulated Laplaces equation, and pioneered the Laplace vary which appears in many branches of mathematical physics. * Laplacian differential operator, widely used in mathematics, is also named after him. * He restated and developed the nebular hypothesis of the origin of the solar system * Was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse. * Laplace made the non-trivial extension of the result to three dimensions to yield a more general set of functions, the spherical harmonics or Laplace coeff icients. Issued his Theorie analytique des probabilites in which he laid down many fundamental results in statistics. * Laplaces most important work was his Celestial Mechanics published in 5 volumes between 1798-1827. In it he sought to give a complete mathematical explanation of the solar system. * In Inductive probability, Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognise as Bayesian. He begins the text with a series of principles of probability, the first six being 1.Probability is the ratio of the favored events to the total possible events. 2. The first principle assumes equal probabilities for all events. When this is not true, we must first determine the probabilities of each event. Then, the probability is the sum of the probabilities of all possible favored events. 3. For independent events, the probability of the occurrence of all is the probability of each compute together. 4. For events not independent, t he probability of event B following event A (or event A causing B) is the probability of A multiplied by the probability that A and B both occur. 5.The probability that A will occur, given that B has occurred, is the probability of A and B occurring divided by the probability of B. 6. Three corollaries are given for the sixth principle, which amount to Bayesian probability. Where event Ai ? A1, A2, An exhausts the list of possible causes for event B, Pr(B) = Pr(A1, A2, An). Then * Amongst the other discoveries of Laplace in pure and applied mathematics are 1. Discussion, contemporaneously with Alexandre-Theophile Vandermonde, of the general theory of determinants, (1772) 2. Proof that every equation of an even degree must have at least one real quadratic factor 3.Solution of the linear partial differential equation of the second order 4. He was the first to consider the difficult problems involved in equations of mixed differences, and to prove that the solution of an equation in finite differences of the first degree and the second order might always be obtained in the form of a continued fraction and 5. In his theory of probabilities 6. Evaluation of several common definite integrals and 7. General proof of the Lagrange reversion theorem. Adrian Marie Legendere Birthdate 18 September 1752 Died 10 January 1833 Nationality French Contributions Well-known and important concepts such as the Legendre polynomials. * He developed the least squares method, which has broad application in linear regression, head processing, statistics, and curve fitting this was published in 1806. * He made substantial contributions to statistics, number theory, abstract algebra, and mathematical analysis. * In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss in connection to this, the Legendre symbol is named after him. * He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. Best kno wn as the author of Elements de geometrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years. * He introduced what are now known as Legendre functions, solutions to Legendres differential equation, used to determine, via power series, the attraction of an ellipsoid at any exterior point. * Published books 1. Elements de geometrie, textbook 1794 2. Essai sur la Theorie des Nombres 1798 3. Nouvelles Methodes pour la Determination des Orbites des Cometes, 1806 4. Exercices de Calcul Integral, book in three volumes 1811, 1817, and 1819 5.Traite des Fonctions Elliptiques, book in three volumes 1825, 1826, and 1830 Simon Dennis Poison Birthdate 21 June 1781 Died 25 April 1840 Nationality French Contributions * He published two memoirs, one on Etienne Bezouts method of elimination, the other on the number of integrals of a finite difference equation. * Poissons well-known correction of Laplaces second order partial differential equation for p otential today named after him Poissons equation or the potential theory equation, was first published in the Bulletin de la societe philomatique (1813). Poissons equation for the divergence of the gradient of a scalar field, ? in 3-dimensional space Charles Babbage Birthdate 26 December 1791 Death 18 October 1871 Nationality English Contributions * Mechanical engineer who originated the concept of a programmable computer. * Credited with inventing the first mechanical computer that eventually led to more complex designs. * He invented the Difference Engine that could compute simple calculations, like multiplication or addition, but its most important trait was its ability create tables of the results of up to seven-degree polynomial functions. Invented the Analytical Engine, and it was the first machine ever designed with the idea of programming a computer that could understand commands and could be programmed much like a present-day(a) computer. * He produced a Table of logarithm s of the natural numbers from 1 to 108000 which was a standard reference from 1827 through the end of the century. Favorite Mathematician Noticeably, Leonard Euler made a mark in the field of Mathematics as he contributed several concepts and formulas that encompasses many areas of Mathematics-Geometry, Calculus, Trigonometry and etc.He deserves to be praised for doing such great things in Mathematics, indeed, his work laid foundation to make the lives of the following generation sublime, ergo, He is my favourite mathematician. VII. Mathematicians in the 19th Century Carl Friedrich Gauss Birthdate 30 April 1777 Died 23 February 1855 Nationality German Contributions * He became the first to prove the quadratic reciprocity law. * Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among things, introduced the symbol ? or congruence and used it in a blanched presentation of modular arithme tic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. * He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer (magnetospheric) sources of Earths magnetic field.Agustin Cauchy Birthdate 21 August 1789 Died 23 May 1857 Nationality French Contributions * His most notable research was in the theory of residues, the question of convergence, differential equations, theory of functions, the legitimate use of imaginary numbers, operations with determinants, the theory of equations, the theory of probability, and the applications of mathematics to physics. * His writings introduced new standards of ri gor in calculus from which grew the modern field of analysis.In Cours danalyse de lEcole Polytechnique (1821), by developing the concepts of limits and continuity, he provided the foundation for calculus essentially as it is today. * He introduced the epsilon-delta definition for limits (epsilon for error and delta for difference). * He transformed the theory of complex functions by discovering integral theorems and introducing the calculus of residues. * Cauchy founded the modern theory of crack by applying the notion of pressure on a plane, and assuming that this pressure was no longer perpendicular to the plane upon which it acts in an elastic body.In this way, he introduced the concept of stress into the theory of elasticity. * He also examined the possible deformations of an elastic body and introduced the notion of strain. * One of the most prolific mathematicians of all time, he produced 789 mathematics papers, including 500 after the age of fifty. * He had sixteen concepts and theorems named for him, including the Cauchy integral theorem, the Cauchy-Schwartz inequality, Cauchy sequence and Cauchy-Riemann equations. He defined continuity in terms of infinitesimals and gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. * He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. * He was the first to define complex numbers as pairs of real numbers. * Most famous for his single-handed development of complex function theory.The first pivotal theorem proved by Cauchy, now known as Cauchys integral theorem, was the following where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. * He was the first to prove Taylors theorem rigorously. * His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced these ar e mainly embodied in his three great treatises 1. Cours danalyse de lEcole royale polytechnique (1821) 2. Le Calcul infinitesimal (1823) 3.Lecons sur les applications de calcul infinitesimal La geometrie (18261828) Nicolai Ivanovich Lobachevsky Birthdate December 1, 1792 Died February 24, 1856 Nationality Russian Contributions * Lobachevskys great contribution to the development of modern mathematics begins with the fifth postulate (sometimes referred to as axiom XI) in Euclids Elements. A modern version of this postulate reads Through a point lying outside a given line only one line can be drawn parallel to the given line. * Lobachevskys geometry found application in the theory of complex numbers, the theory of vectors, and the theory of relativity. Lobachevskiis deductions produced a geometry, which he called imaginary, that was internally consistent and harmonious yet different from the traditional one of Euclid. In 1826, he presented the paper Brief Exposition of the Principles of Geometry with Vigorous Proofs of the Theorem of Parallels. He refined his imaginary geometry in subsequent works, dating from 1835 to 1855, the last being Pangeometry. * He was well respected in the work he developed with the theory of infinite series especially trigonometric series, integral calculus, and probability. In 1834 he found a method for approximating the roots of an algebraic equation. * Lobachevsky also gave the definition of a function as a correspondence between two sets of real numbers. Johann Peter Gustav Le Jeune Dirichlet Birthdate 13 February 1805 Died 5 May 1859 Nationality German Contributions * German mathematician with belatedly contributions to number theory (including creating the field of analytic number theory) and to the theory of Fourier series and other topics in mathematical analysis. * He is credited with being one of the first mathematicians to give the modern formal definition of a function. Published important contributions to the biquadratic reciprocity law. * In 1837 he published Dirichlets theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. * He introduced the Dirichlet characters and L-functions. * In a couple of papers in 1838 and 1839 he proved the first class number formula, for quadratic forms. * Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlets approximation theorem. * In 1826, Dirichlet proved that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. * Developed significant theorems in the areas of elliptic functions and applied analytic techniques to mathematical theory that resulted i n the fundamental development of number theory. * His lectures on the equilibrium of systems and potential theory led to what is known as the Dirichlet problem.It involves finding solutions to differential equations for a given set of values of the bounds points of the region on which the equations are defined. The problem is also known as the first boundary-value problem of potential theorem. Evariste Galois Birthdate 25 October 1811 Death 31 May 1832 Nationality French Contributions * His work laid the foundations for Galois Theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. * He was the first to use the word group (French groupe) as a technical term in mathematics to represent a group of permutations. Galois published three papers, one of which laid the foundations for Galois Theory. The second one was about the numerical resolution of equations (root finding in modern terminology). The third was an important one in number th eory, in which the concept of a finite field was first articulated. * Galois mathematical contributions were published in full in 1843 when Liouville reviewed his disseminated sclerosis and declared it sound. It was finally published in the OctoberNovember 1846 issue of the Journal de Mathematiques Pures et Appliquees. 16 The most famous contribution of this manuscript was a novel proof that there is no quintic formula that is, that fifth and higher degree equations are not generally solvable by radicals. * He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today. * One of the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. * Galois most significant contribution to mathematics by far is his development of Galois Theory.He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois orig