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  • 1.   The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers  or a sixmonth lunar calendar. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10.“  The Ishango bone, according to scholar  Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this, however, is disputed. Predynastic Egyptians of the 5th millennium BC pictorially represented  geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.
  • 2.   The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek  (mahatma), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.Chinese mathematics  made early contributions, including a place value system.The  Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in Indiaand was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an  increasing pacethat continues through the present day.
  • 3. ************************* *************************     Archimedes was a Greek mathematician, philosopher and inventor who wrote important works on geometry, arithmetic and mechanics. Archimedes was born in Syracuse on the eastern coast of Sicily and educated in Alexandria in Egypt. He then returned to Syracuse, where he spent most of the rest of his life, devoting his time to research and experimentation in many fields. In mechanics he defined the principle of the lever and is credited with inventing the compound pulley and the hydraulic screw for raising water from a lower to higher level. He is most famous for discovering the law of hydrostatics, sometimes known as 'Archimedes' principle', stating that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. Archimedes is supposed to have made this discovery when stepping into his bath, causing him to exclaim 'Eureka!' During the Roman conquest of Sicily in 214 BC Archimedes worked for the state, and several of his mechanical devices were employed in the defence of Syracuse. Among the war machines attributed to him are the catapult and - perhaps legendary - a mirror system for focusing the sun's rays on the invaders' boats and igniting them. After Syracuse was captured, Archimedes was killed by a Roman soldier. It is said that he was so absorbed in his calculations he told his killer not to disturb him.
  • 4. ***********************  Leonhard Euler 15 April 1707 – 18 September 1783) was a pioneering Swiss  mathematician and physicist. 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.[3] He is also renowned for his work in  mechanics, fluid dynamics,  optics, and astronomy.]
  • 5.  Euler spent most of his adult life in St. Petersburg, Russia, and inBerlin, Prussia. He is considered to be the pre-eminent mathematician of the 18th century, and one of the greatest mathematicians ever. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes.[4] A statement attributed to Pierre-Simon Laplaceexpresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."[5] Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor  of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the  Bernoulli family—Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies ofDescartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.[6] Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono.[7] At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, a man who became known as "the father of naval architecture" won, and Euler took second place. Euler later won this annual prize twelve times.[8
  • 6.     Johann Carl Friedrich Gauss  (30 April 1777 – 23 February 1855) was a German mathematician and  physical scientist who contributed significantly to many fields, including number theory, algebra,  statistics,analysis,  differential geometry, geodesy,  geophysics, electrostatics,  astronomy and optics. Sometimes referred to as the Princeps mathematicorum("the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.[2]
  • 7.     Carl Friedrich Gauss was born on 30 April 1777 in  Lower Saxony, Germany, as the son of poor working-class parents. Indeed, his mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which itself occurs 40 days after  Easter. Gauss would later solve this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years. He was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. There are many anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completedDisquisitiones Arithmeticae, his  magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick,who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the  University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.He further advanced modular arithmetic, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The  prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbersare distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three triangular numberson 10 July and then jotted down in his diary the famous note: "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.
  • 8.   French mathematician who did important work in many different branches of mathematics. However, he did not stay in any one field long enough to round out his work. He had an amazing memory and could state the page and line of any item in a text he had read. He retained this memory all his life. He also remembered verbatim by ear. His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper. Despite his keen mathematical ability, he was physically clumsy and artistically inept. In fact, he received a score of 0 on his Polytechnique entrance exam. He was always in a rush and disliked going back for changes or corrections. He was also a popularizer of mathematics. Poincaré's brother Raymond was president of the French Republic during World War I. Poincaré is quoted as saying, "It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed" (Boyer and Merzbach 1991, p. 599). In 1880, he created generalized  elliptic functions  called automorphic functions.  He discovered that automorphic functions invariant under the same group are connected by an algebraic equation. Conversely, he found that the coordinates of a point on any algebraic curve  can be expressed in terms of automorphic functions.  He showed they could be used to solve second order linear differential equation  with algebraic coefficients.
  • 9.    As a mathematician and physicist, After receiving his degree, Poincaré began teaching at the University of Caen in Normandy (in December 1879). At the same time he published his first major article – they are devoted to treatment with a class of automorphic functions. There, in Caen, he met his future wife, Louise Poulin d'Andesi and on April 20, 1881, they held their wedding. They had a son and three daughters Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. During the years of 1883 to 1897, he taught mathematical analysis in École Polytechnique. In 1881–1882, Poincaré created a new branch of mathematics: the qualitative theory of differential equations. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in  celestial mechanics and mathematical physics.
  • 10.    David Hilbert ( January 23, 1862 – February 14, 1943) was aGerman mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces,[3] one of the foundations of functional analysis. Hilbert adopted and warmly defended  Georg Cantor's set theory and  transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
  • 11. DAVID HILBERT   Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked at the time of his birth. In the fall of 1872, he entered the Friedrichskolleg  Gymnasium (Collegium fridericianum, the same school thatImmanuel Kant had attended 140 years before), but after an unhappy period he transferred to (fall 1879) and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium. Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the "Albertina". In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters),returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski."[ In 1884, Adolf Hurwitzarrived from Göttingen as an Extraordinarius, i.e. an associate professor. An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special  binary forms, in particular the spherical harmonic functions"). Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own“. at Königsberg they had their one child, Franz Hilbert (1893–1969). In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world. He remained there for the rest of his life.
  • 12.     His son Franz suffered throughout his life from an undiagnosed mental illness: his inferior intellect was a terrible disappointment to his father and this misfortune was a matter of distress to the mathematicians and students at Göttingen.Minkowski — Hilbert's "best and truest friend"— died prematurely of a ruptured appendix in 1909. The Mathematical Institute in Göttingen. Its new building, constructed with funds from the  Rockefeller Foundation, was opened by Hilbert and Courant in 1930. The Göttingen school among the students of Hilbert were Hermann Weyl, chess champion  Emanuel Lasker,Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church. Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus  (1911), and Wilhelm Ackermann (1925).Between 1902 and 1939 Hilbert was editor of the  Mathematische Annalen, the leading mathematical journal of the time.
  • 13. ************************* ************************   Euclid published a great number of works on a variety of topics, but is often referred to as the father of geometry and is most remembered for the thirteenvolume textbook Elements. Primarily composed of the accumulated knowledge of other mathematicians, Elements contains a few original theorems that are directly attributed to Euclid and are the foundation of his fame as a brilliant mathematician. Euclid is also generally praised for the clarity and logic with which he systemically presents geometric principles, theorems, and proofs, within Elements, causing it to remain a standard mathematical text for over two thousand years. Though often overshadowed by his mathematical reputation, Euclid is a central figure in the history of optics. He wrote an in-depth study of the phenomenon of visible light inOptica, the earliest surviving treatise concerning optics and light in the western world. Within the work, Euclid maintains the Platonic tradition that vision is caused by rays that emanate from the eye, but also offers an analysis of the eye's perception of distant objects and defines the laws of reflection of light from smooth surfaces. Optica was considered to be of particular importance to astronomy and was often included as part of a compendium of early Greek works in the field. Translated into Latin by a number of writers during the medieval period, the work gained renewed relevance in the fifteenth century when it underpinned the principles of linear perspective.
  • 14. ************************* ************************   In mathematics, the Pythagorean theorem—or Pythagoras' theorem—is a relation in  Euclidean geometry among the three sides of a right triangle. It states that the square of the  hypotenuse (the side opposite the  right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an  equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1] where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
  • 15. ************************ *************************    John Napier of Merchiston (1550 – 4 April 1617) – also signed as Neper, Nepair – namedMarvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, andastronomer. He was the 8th Laird of  Merchistoun. John Napier is best known as the inventor of logarithms. He also invented the so-called "Napier's bones" and made common the use of the decimal point in arithmetic and mathematics. Napier's birthplace, Merchiston Tower  in Edinburgh, Scotland, is now part of the facilities of Edinburgh Napier University. After his death from the effects of gout, Napier's remains were buried in  St Cuthbert's Church, Edinburgh.
  • 16. ************************* *************************      Aryabhatta is the author of several treatises on mathematics  and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic,  algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. The Arya-siddhanta, a lot work on astronomical computations, is known through the writings of Aryabhat ta's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta  and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon, a shadow , possibly anglemeasuring devices, semicircular and circular cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical. A third text, which may have survived in the Arabic  translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the  Persian scholar and chronicler of India, 
  • 17. ************************* ************************  Varahamihir's main work is the book Pañcasiddhāntikā  on the Five [Astronomical] Canons) dated ca. 575 CE gives us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya Siddhanta, Romaka Siddhanta,  Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhantas. It is a compendium of Vedanga Jyotisha as well as Hellenistic astronomy(including Greek, Egyptian and Roman elements).He was the first one to mention in his work Pancha Siddhantika that the ayanamsa, or the shifting of the equinox is 50.32 seconds.6655
  • 18. *************************** *************************   Brahmagupta (Sanskrit: ब्रह्मगुप;  listen ( help·info)) (597–668 AD) was an Indian  mathematician andastronomer who wrote two important works on mathematics and astronomy: theBrāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma) (628), a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text. There are reasons to believe that Brahmagupta originated from Bhinmal. Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.
  • 19.     Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta, The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. Therupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted which is a solution for the equation  equivalent to , where rupas refers to the constants c and e. He further gave two equivalent solutions to the general quadratic equation 18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number]. 18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.