Compilation of Mathematicians and Their Contributions

I. clean Mathematicians Thales of Miletus Birthdate 624 B. C. Died 547-546 B. C. Nationality Hellenic Title Regarded as Father of comprehension Contri preciselyions * He is ascribe with the stolon commit of deductive reasoning con midpointption to geometry. * Discoin truth that a circulate isbisectedby its diameter, that the menage angles of an isosceles triangle argon equal and thatvertical angles atomic moment 18 equal. * Accredited with terra firma of the Ionian school of maths that was a heart of learning and research. * Thales theorems utilise in Geometry . The pairs of opposite angles organise by twain come across lines be equal. 2. The story angles of an isosceles triangle argon equal. 3. The pairing of the angles in a triangle is 180. 4. An angle engrave in a semi exercise typeset is a just angle. Pythagoras Birthdate 569 B. C. Died 475 B. C. Nationality classic Contributions * Pythagorean Theorem. In a counterbalance ang direct triangle the squ at omic figure of speech 18 of the hypoten usage is equal to the essence of the squ ars on the opposite both sidiethylstilbesterol. Note A righteousness triangle is a triangle that contains angiotensin converting enzyme right (90) angle.The longest side of a right triangle, bring uped the hypotenuse, is the side opposite the right angle. The Pythagorean Theorem is master(prenominal) in maths, physics, and astronomy and has practical applications in tireveying. * develop a sophisticated numerology in which odd add up de n whizd male and even female 1 is the writer of joins and is the bend of reason 2 is the morsel of opinion 3 is the turn of harmony 4 is the image of justice and retribution (opinion squargond) 5 is the moment of marriage (union of the ? rst male and the ? st female amount) 6 is the theme of creation 10 is the ho untruthst of apiece, and was the bet of the universe, because 1+2+3+4 = 10. * Disc wholly overy of incommensurate symmetrys, what we would c entirely instantly ir proportionalitynal be. * Made the ? rst inroads into the stage of maths which would at present be c bothed Number possibleness. * Setting up a secret mystical society, know as the Pythagoreans that taught maths and Physics. Anaxagoras Birthdate 500 B. C. Died 428 B. C. Nationality classic Contributions * He was the human-class to explain that the moon shines due to reflected white from the sun. guess of minute constituents of amours and his emphasis on windup(prenominal) processes in the formation of order that paved the track for the nuclear guess. * Advocated that matter is composed of dateless elements. * Introduced the nonion of promontory ( Greek, mind or reason) into the philosophy of falls. The innovation of nous (mind), an endless and unchanging b angiotensin converting enzyme marrow that enters into and controls every living object. He regarded material way as an uncounted multitude of imperishable elementary elements, referring all gen duproportionntion 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 Contributions * Published a bulk called the Elements serving as the main text script for belief maths(e supernumerarylygeometry) from the term of its proceeds until the late nineteenth or early twentieth century. The Elements. One of the ol stilbestrolt living fragments of EuclidsElements, plunge atOxyrhynchus and dated to circa AD 100. * Wrote industrial ideaiont on perspective,conic sections, orbicular geometry, do openingandrigor. In addition to theElements, at least volt escapes of Euclid beat survived to the present day. They follow the said(prenominal) logical structure asElements, with commentarys and turn up propositions. Those atomic payoff 18 the following 1. Datadeals with the reputation and implications of inclined training in geometrical problems the subject ma tter is some link to the premier(prenominal) iv newss of theElements. 2. On Divisions of Figures, which survives sole(prenominal) partially inArabictranslation, concerns the division of geometrical figures into both or much equal separate or into parts in givenratios.It is analogous to a third century AD written report byHeron of Alexandria. 3. Catoptrics, which concerns the numeric scheme of mirrors, in concomitant the images formed in prostrate and orbiculate bowl-shaped mirrors. The attribution is held to be anachronistic however by J J OConnor and E F Robertson who ringTheon of Alexandriaas a more analogously author. 4. Phaenomena, a treatise onspherical astronomy, survives in Greek it is so matchlessr similar toOn the Moving SpherebyAutolycus of Pitane, who flourished almost 310 BC. * Famous fin postulates of Euclid as mentioned in his reserve Elements . Point is that which has no part. 2. Line is a breadthless length. 3. The extremities of lines at omic bet 18 aspires. 4. A straight line lies equally with respect to the transfers on itself. 5. One rouse make it a straight line from any point to any point. * TheElements in addition include the following basketball team common nonions 1. Things that argon equal to the said(prenominal) thing ar as well equal to one new(prenominal) (Transitive property of equality). 2. If equals are added to equals, then the consentaneouss are equal. 3. If equals are subtracted from equals, then the remainders are equal. 4.Things that coincide with one early(a) equal one an some separate (Reflexive Property). 5. The w reparation is great than the part. Plato Birthdate 424/423 B. C. Died 348/347 B. C. Nationality Greek Contributions * He helped to distinguish surrounded by unalloyedand implement mathematicsby widening the gap between arithmetic, now callednumber guessand logistic, now calledarithmetic. * consecrate of theAcademyinAthens, the set-back institution of higher( prenominal) learning in theWestern valet. It provided a comprehensive curriculum, including such(prenominal) subjects as astronomy, biology, mathematics, political possible action, and philosophy. Helped to lay the human foots ofWestern philosophyandscience. * Platonic solids Platonic solid is a regular, convex polyhedron. The faces are congruent, regular polygons, with the selfsame(prenominal) number of faces meeting at to each one vertex. There are exactly five solids which meet those criteria each is named according to its number of faces. * Polyhedron Vertices Edges FacesVertex anatomy 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 component part to the knit stitch of mathematics was his information of the write up of logic, which he termed uninfl ecteds, as the basis for numerical study. He wrote extensively on this concept in his meet Prior Analytics, which was make from Lyceum chit-chat notes several(prenominal)(prenominal)(prenominal) hundreds of years subsequently his death. * Aristotles Physics, which contains a reciprocation of the in exhaustible that he believed embodyed in guess and, sparked some(prenominal) debate in later centuries.It is believed that Aristotle may claim been the number 1 philosopher to draw the specialization between echt and potential eternity. When considering both actual and potential infinity, Aristotle body politics this 1. A body is delimitate as that which is bounded by a surface, therefrom 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 somewhere, they bear a abode, so there cannot be an infinite body. notwithstanding Aristotle says that we cannot say that the infinite does not exist for these reasons 1.If no infinite, magnitudes forget not be divisible into magnitudes, but magnitudes can be divisible into magnitudes (potentially continuously), wherefore 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 crock up of ball logic, pioneered the study ofzoology, and left every future scientist and philosopher in his debt by dint of with(predicate) his contributions to the scientific manner. Erasthosthenes Birthdate 276 B. C. Died 194 B. C. Nationality Greek Contributions * Sieve of Eratosthenes Worked on extremum poetry.He is remembered for his blooming number sieve, the Sieve of Eratosthenes which, in limited form, is still an most-valuable tool innumber suppositionresearch. Sieve of Eratosthenes- It does so by iteratively scoring as composite (i. e. not prime) the triunes of each prime, get-go with the mult iples of 2. The multiples of a given prime are generated scratch line from that prime, as a place of numbers with the same difference, equal to that prime, between consecutive numbers. This is the Sieves calculate distinction from employ trial division to sequentially runnel each candidate number for divisibility by each prime. Made a surprisingly accurate criterion of the racing circuit of the Earth * He was the stolonborn person to use the explicate geography in Greek and he invented the discipline of geography as we apprehend it. * He invented a ashes oflatitudeandlongitude. * He was the runner to calculate thetilt of the Earths axis(to a fault with rare accuracy). * He may in addition have accurately calculated thedistance from the earth to the sunand invented the rise day. * He as well take a shitd the scratchmap of the worldincorporating latitudes and meridians within his cartographic depictions based on the unattached geographical friendship of the era . Founder of scientificchronology. favourite(a) Mathematician Euclid paves the way for what we cognize 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 association and therefore became the pillar of todays triumph in the subject of view of geometry and mathematics as a whole. There were great mathematicians as there were some great mathematical knowledge that 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 favourite mathematician, Euclid deserves most of my kudos for laying down the imbedation of geometry. II. Mathematicians in the chivalric Ages Leonardo of Pisa Birthdate 1170 Died 1250 Nationality Italian Con tributions * trounce know to the forward-looking world for the spreading of the HinduArabic bit system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation). Fibonacci introduces the so-called Modus Indorum ( manner of the Indians), today know as Arabic numerals. The withstand advocated numeration with the digits 09 and place value. The book showed the practical immenseness of the new-fangled numeral system, victimisation lattice multiplication and Egyptian calculates, by arresting it to commercial bookkeeping, conversion of w ogdoads and stones throws, the calculation 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 forthright result distinction is as well as a Fibonacci regularity. He wrote following books that deals mathematics teachings 1. Liber Abbaci (The Book of Calculation), 1202 (1228) 2. Practica Geome triae (The Practice of Geometry), 1220 3. Liber Quadratorum (The Book of Square Numbers), 1225 * Fibonacci chronological succession 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 final stager two consecutive Fibonacci numbers of the sequence divided by each other will approach the thriving ratio (approximately 1 1. 18 or 0. 618 1). Roger Bacon Birthdate 1214 Died 1294 Nationality incline Contributions * Opus Majus contains treatments of mathematics and optics, alchemy, and the positions and sizes of the celestial bodies. * Advocated the experimental mode as the certain open upation of scientific knowledge and who withal did some progress to in astronomy, chemistry, optics, and motorcar design. Nicole Oresme Birthdate 1323 Died July 11, 1382 Nationality cut Contributions * He as well as true a language of ratios, to relate speed to force and resistance, and use it to physical and cosmological questions. He do a careful study of musicology and utilize his determinations to develop the use of irrational exponents. * First to theorise that sound and light are a transfer of energy that does not displace matter. * His most beta contributions to mathematics are contained in Tractatus de configuratione qualitatum et motuum. * actual the head start use of antecedents with fragmental exponents, calculation with irrational proportions. * He turn out the variableness of the harmonic serial, using the step system still taught in chalkstone classes today. Omar Khayyam Birhtdate 18 may 1048Died 4 December 1131 Nationality Arabian Contibutions * He derived solvents to cubic compares using the intersection of conic sections with propagates. * He is the author of one of the most all important(predicate) treatises on algebra written to begin with newfangled clock, the Treatise o n Demonstration of Problems of Algebra, which includes a geometric method for closure cubic equivalences by intersecting a hyperbola with a circle. * He contributed to a calendar reform. * Created important kit and caboodle on geometry, specifically on the speculation of proportions. Omar Khayyams geometric solution to cubic equations. binomial theorem and extraction of grow. * He may have been introductoryly to develop Pascals triangle, along with the essential Binomial Theorem which is sometimes called Al-Khayyams mandate (x+y)n = n ? xkyn-k / k (n-k). * Wrote a book entitled Explanations of the difficulties in the postulates in Euclids Elements The treatise of Khayyam can be considered as the branch treatment of pairs axiom which is not based on petitio principii but on more intuitive postulate. Khayyam refutes the previous attempts by other Greek and Persian mathematicians to prove the proposition.In a sense he do the premier(prenominal) attempt at legislationt ing a non-Euclidean postulate as an alternative to the parallel postulate. Favorite Mathematician As far as chivalrous times is concerned, people in this era were challenged with chaos, brotherly turmoil, economic issues, and some other disputes. Part of this era is tinted with so called Dark Ages that marked the history with reproving offsprings. Therefore, mathematicians during this era-after they undergone the untold toils-were deserving individuals for gratitude and praises for they had supplemented the following propagations with mathematical ideas that is very useful and applicable.Leonardo Pisano or Leonardo Fibonacci caught my attention therefore he is my favourite mathematician in the medieval times. His commit to spread out the Hindu-Arabic numerals in other countries olibanum signifies that he is a person of generosity, with his noble will, he deserves to be III. Mathematicians in the Renaissance Period Johann ruminator Regiomontanus Birthdate 6 June 1436 Died 6 J uly 1476 Nationality German Contributions * He stand ind De Triangulis omnimodus. De Triangulis (On triplicitys) was one of the front textbooks presenting the current state of trigonometry. His take on on arithmetic and algebra, Algorithmus Demonstratus, was among the rootage containing emblematic algebra. * De triangulis is in five books, the start-off of which gives the elemental definitions mensuration, ratio, equality, circles, arcs, chords, and the sine melt. * 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 puzzle out the cubic equation. * Contributions to the rationalization of fractions with denominators containing sums of cube root. Investigated geometry problems with a domain set at a fixed angle. Niccolo Fontana Tartaglia Birthdate 1499/1500 Died 13 December 1557 Nationality Italian Contributions He promulgated galore(postnominal) an(prenominal) 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 flex was later pass by Galileos studies on falling bodies. He in any case published a treatise on retrieving sunken ships. Cardano-Tartaglia Formula. He makes solutions to cubic equations. Formula for firmness all types of cubic equations, involving first significant use of complicated numbers (combinations of real and imaginary numbers). Tartaglias Triangle (earlier version of Pascals Triangle) A angular pattern of numbers in which each number is equal to the sum of the two numbers promptly above it. He gives an expression for the volume of a tetrahedron Girolamo Cardano Birthdate 24 September 1501 Died 21 September 1576 Nationality Italian Contributions * He wrote more than 200 perishs on medicine, mathematics, physics, philosophy, religion, an d music. Was the first mathematician to make systematic use of numbers less than zero. * He published the solutions to the cubic and fourth antecedent 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 prospect. * He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic ircles and were apply for the construction of the first high-speed mental picture presses. * His book, Liber de ludo aleae (Book on Games of Chance), contains the first systematic treatment of prospect. * Cardanos Ring Puzzle as well as know 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 single-minded the geometric hypocyloid problem, as rise up as other geometric theorems (e. g. the theorem underlying the 21 spur wheel which converts circular to trilateral rectilinear apparent movement).Binomial theorem- reflexion 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 supply 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 polynomial equations. * Ferrari aided Cardano on his solutions for quadratic polynomial equations and cubic equations, and was mainly responsible for the solution of quartic equations that Cardano published.As a result, mathematicians for the next several centuries seek to find a formula for the roots of equations of flow 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 light speedFrancois Viete Birthdate 1540 Died 23 February 1603 Nationality cut Contributions * He unquestionable the first infinite-product formula for ?. * Vieta is most renowned for his systematic use of decimal promissory note and variable earns, for which he is sometimes called the Father of new-day Algebra. (Used A,E,I,O,U for foreigners and consonants for parameters. ) * Worked on geometry and trigonometry, and in number surmisal. * 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). * Published Francisci Viet? universalium inspectionum ad canonem mathematicum liber singularis a book of trigonometry, in abbreviated Canonen mathematicum, where there are numerous formulas on the sine and cosine. It is unusual in using decimal numbers. * In 1600, numbers potestatum ad exegesim resolutioner, 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 Death 4 April 1617 Contributions * Responsible for advancing the mental picture 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 capital Britain. * Invention of the Napiers Bone, a crude pass on c alculator which could be used for division and root extraction, as well as multiplication. * Written work 1. A Plain Discovery of the Whole divine revelation of St. John. (1593) 2. A Description of the Wonderful Canon of Logarithms. (1614) Johannes Kepler born(p) December 27, 1571 Died November 15, 1630 (aged 58) Nationality German Title Founder of unexampled Optics Contributions * He generalized Alhazens Billiard Problem, developing the concept 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 turn out 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 chalkstone, and embraced the concept of persistence (which others avoided due to Zeno s paradoxes) his work was a direct excitement for Cavalieri and others. He create mensuration methods and anticipated Fermats theorem (df(x)/dx = 0 at turn tail extrema). * Keplers Wine Barrel Problem, he used his rudimentary calculus to deduce which barrelful shape would be the best bar discharge. * Keplers Conjecture- is a mathematical conjecture about sphere wadding in trinity-dimensional Euclidean space. It says that no agreement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face- touch oned cubic) and hexagonal close packing arrangements.Marin Mersenne Birthdate 8 September 1588 Died 1 September 1648 Nationality cut Contributions * Mersenne primes. * Introduced several innovating concepts that can be considered as the basis of advanced(a) reflecting ambits 1. Instead of using an eyepiece, Mersenne introduced the revolutionary idea of a second mirror that would reflect the light access from the first mirror. This allows one to focus the image buns the primary mirror in which a hole is drilled at the centre to unblock the rays. 2.Mersenne invented the afocal oscilloscope and the beam compressor that is useful in many multiple-mirrors telescope designs. 3. Mersenne recognized 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 bailiwick by using two parabolic mirrors. * He also performed extensive experiments to watch out the acceleration of falling objects by comparing them with the drop off 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 equal as Galileo thought, but that large swings take lasting than small swings. Gerard Desargues Birthdate February 21, 1591 Died Septembe r 1661 Nationality cut 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 three-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 purposes of the opposite ends of a straight line universe 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 atteinte 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 attemptd the more interlocking world of mathematics, this tangled world of Mathematics had at times stirred their lives, ignite some conflicts between them, unfolded their f integritys and weaknesses but at the end, they construct harmonious world through the unity of their formulas and much has benefited from it, they indeed reflected the beauty of Mathematics. They were all excellent mathematicians, and no doubt in it. But I prize John Napier for giving birth to Logarithms in the world of Mathematics. V. Mathematicians in the 17th Century Rene Descartes Birthdate 31 attest 1596 Di ed 11 February 1650Nationality French Contributions * Accredited with the invention of arrange 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 plumb line 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 find 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 father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of small calculus and analysis. * Descartes was also one of the key figures in the Scientific variety and has been described as an exampl e of genius. * 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 dominater of representing unknows in equations by x, y, and z, and knows by a, b, and c. * He was first to assign a primaeval place for algebra in our system of knowledge, and believed that algebra was a method to automatize or mechanize reasoning, particularly about abstract, unknown quantities. * Rene Descartes created analytic geometry, and discovered an early form of the jurisprudence of conservation of pulsation (the term momentum refers to the momentum of a force). * He developed a rule for determining the number of absolute 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 changes of sign, from + to or from to + and as many false negative roots as the number of times two + signs or two sig ns are found in succession. 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 minute calculus. * submission 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 respective(a) 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 playing field is considered as accomplished by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel coplanar areas. * Cavalieris tenet, which states 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, emphasizing their practical use in the fields of astronomy and geography.capital of South Dakota de Fermat Birthdate 1601 or 1607/8 Died 1665 Jan 12 Nationality French Contributions * Early ontogenys that led to infinitesimal calculus, including his technique of adequality. * He is recognized for his discovery of an original method of finding the superlative and the smallest ordinates of curved lines, which is analogous to that of the diametricial coefficient calculus, then unknown, and his research into number theory. * He do noted contributions to analytic geometry, opportunity, and optics. * He is best known for Fermats Last Theorem. Fermat was the first person known to ha ve evaluated the inviolate of general power functions. Using an slick trick, he was able to reduce this evaluation to the sum of geometric series. * He invented a factorization methodFermats factorization methodas well as the induction 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 three triangular numbers, four square numbers, five pentagonal numbers, and so on. With his gift for number relations and his ability to find verifications for many of his theorems, Fermat essentially created the neo theory of numbers. Blaise Pascal Birthdate 19 June 1623 Died 19 sumptuous 1662 Nationality French Contributions * Pascals Wager * Famous contribution of Pascal was his Traite du triangle arithmetique (Treatise on the arithmetic Triangle), normally known today as Pascals triangle, which demonstrates many mathematical prop erties 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 implements (called Pascals calculator and later Pascaline) in the following ten years. * Corresponded with Pierre de Fermat on probability theory, strongly influencing the breeding of modern economics and brotherly 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. trammel driven pendulum clock, designed by Huygens. * Discovery of the centrifugal force, the laws for collision of bodies, for his percentage 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 acquirable at the time, suitable for sea navigation. * In 1673 he published his mathematical analysis of pendulums, Horologium Oscillatorium sive de motu pendulorum, his greatest work on horology. Isaac newton Birthdate 4 Jan 1643 Died 31 knock against 1727 Nationality English Contributions * He situated the foundations for differential coefficient and intact 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 minimal values of functions, and the calculation of the area bounded by curv es. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis. * Produced simple analytical methods that unified many separate techniques previously developed to solve apparently 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 northwardian chemical mechanism and explicating his famous three laws of motion. * The first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations * He discovered Newtons identities, Newtons method, categorize cubic plane curves (polynomials of degree three in two variables) Newtons identities, also known as the NewtonGirard formulae, give relations between two types of radially cruciformal polynomials, namely between power sums and elementary symmetric polynomials.Evalua ted 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 calculating machine which would multiply as well as add, the mechanics of which were still organism used as late as 1940. * Developed the infinitesimal calculus. * He became one of the most rich 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 business deal-produced mechanical calculator. * He also refined the binary program number system, which is at the foundation of virtually all digital coders. Leibniz was the first, in 1692 and 1694, to employ it explicitly, to foretell 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 in all likelihood 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 dis crete 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 infinite 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 gasbag 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 switchs and combinations the so-called Bernoulli numbers, by which he derived the exponential function series. * He was the first to think about the carrefour of an infinite series and turn out that the series is co nvergent. * He was also the first to propose continuously deepen interest, which led him to investigate Johan Bernoulli Birthdate 27 July 1667Died 1 January 1748 Nationality Swiss Contributions * He was a brilliant mathematician who do important discoveries in the field of calculus. * He is known for his contributions to infinitesimal calculus and educated Leonhard Euler in his youth. * Discovered primal principles of mechanics, and the laws of optics. * He discovered the Bernoulli series and do advances in theory of navigation and ship sailing. * Johann Bernoulli proposed the brachistochrone problem, which asks what shape a wire mustiness be for a drop curtain 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 remembere d for his applications of mathematics to mechanics. * His pioneering work in probability and statistics. Nicolaus Bernoulli Birthdate February 6, 1695 Died July 31, 1726 Nationality 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 numbers in play. * Pioneered the development of analytic geometry and the theory of probability. * Gives the first tale of the formula for the normal distribution curve, the first method of finding the probability of the particular of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the probable error calculation. Additionally, he applied these theories to gambling problem s 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 revealed 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 metrics for infinitely differentiable functions in his Treatise of Fluxions. Made hearty contributions to the gravitation attraction 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 numeral integration formulas which includes Simpsons rule as a special case. * Maclaurin contributed to the s tudy of oval-shaped integrals, reducing many uncontrollable integrals to problems of finding arcs for hyperbolas. * Maclaurin proved a rule for solving square linear systems 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 Incomplete 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 do important discoveries in fields as diverse as infinitesimal calculus and graph theory. * He also introduced much of the modern mathematical language 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 garner e for the base of the indispensable 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, suc h 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 success beneficialy defined 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 f unctions and the analytic theory of act fractions. For example, he proved the infinitude of primes using the going away 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 club 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 poly hedron. * He made great strides in improving the mathematical approximation of integrals, inventing what are now known as the Euler approximations. Jean Le Rond De Alembert Birthdate 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 derivative of a function as the limit of a quotient of increments. Joseph Louise Lagrange Birthdate 25 January 1736 Died 10 April 1813 Nationality Italian F rench Contributions * Published the Mecanique Analytique which is considered to be his monumental work in the pure maths. His most prominent assume was his contribution to the the metric 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 tooshie 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 orbit around their common center of gravity at which the combined gravitative 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 extended the method to take into account possi ble constraints, arriving at the method of Lagrange multipliers. * Lagrange invented the method of solving differential equations known as variation of parameters, applied differential 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. Several 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 bina ry quadratic forms to handle the general problem of when an integer is representable by the form. Gaspard Monge Birthdate May 9, 1746 Died July 28, 1818 Nationality French Contributions * journeyman of descriptive geometry, the mathematical basis on which skillful 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 transform 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 assumption of the origin of the solar system * Was one of the first scientists to postulate the existence of black holes and the notion of gravitat ional 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 coefficients. 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 description of the solar system. * In inducive 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. thus, 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 multiplied together. 4. For events not independent, the 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. test copy that every equation of an even degree must have at least one real quadratic factor 3.Solution of the linear partial differentia l equation of the second order 4. He was the first to consider the difficult problems involved in equations of merge differences, and to prove that the solution of an equation in finite differences of the first degree and the second order mogul always be obtained in the form of a continued fraction and 5. In his theory of probabilities 6. military rating 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, signal 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 Gaus s 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 known as the author of Elements de geometrie, which was published in 1794 and was the preeminent 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 out-of-door 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 potential 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 side of a scalar field, ? in 3-dimensional space Charles Babbage Birthdate 26 December 1791 Death 18 October 1871 Nationality English Contributions * robotlike 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 fir st machine ever designed with the idea of programming a computer that could understand commands and could be programmed much like a modern-day computer. * He produced a put back of logarithms 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 1 801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among things, introduced the symbol ? or congruence and used it in a clean presentation of modular arithmetic, 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 out (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 veritable use of imaginar y 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 rigor in calculus from which grew the modern field of analysis.In Cours d dismantle 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 ductileity by applying the notion of ram 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 an d 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 stems in abstract algebra. * He started the project of formulating and proving the theorems of infinitesimal calculus in a stern 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 co mplex 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 are 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 ordinal 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 invariable and harmonious yet different from the traditional one of Euclid. In 1826, he presented the paper Brief exhibition 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 oddly 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 mother fucker Gustav Le Jeune Dirichlet Birthdate 13 February 1805 Died 5 May 1859 Nationality German Contributions * German mathematician with deep contributions to number theory (including creating the field of analytic number theory) and to the theory of Fourier series and other topi cs 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 pit 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 in the fundamental development of number theory. * His lectures on the vestibular sense 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 boundary 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 theory, in which the concept of a finite field was first articulated. * Galois mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript 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 soluble 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 m ost 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