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1. 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. 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. 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. 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. 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. 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]