4. Background Archimedes of Syracuse (pronounced ar-ka-meed-eez) is considered one of the greatest mathematicians in history. In fact, he is believed to be one of the three greatest mathematicians along with Isaac Newton and Carl Gauss. His greatest contributions to mathematics were in the area of Geometry. Archimedes was also an accomplished engineer and an inventor. He was believed to have been obsessed with Geometry though. Archimedes was born in Syracuse, at that time a self-governing in Magna Graecia, Greece in 287 BC and died 212 BC after being killed by a Roman soldier who did not know who Archimedes was. The date of birth is based on a statement by the historian that Archimedes lived for 75 years. He was the son of an astronomer: Phidias of whom we know nothing about. Archimedes received his formal education in Alexandria, Egypt which at the time was considered to be the 'intellectual center' of the world. When he completed his formal studies in Alexandria, he returned and stayed in Syracuse for the rest of his life. It is not known whether he ever married or had children. Archimedes is known to be a relative of Hiero II, who was the king of Syracuse during Archimedes' life. Hiero and Archimedes were very close friends. However, nothing else is known about any other members of Archimedes' family. Archimedes lived in Syracuse for his whole life, except for when he went to school in Alexandria, and at one point it was up to Archimedes' inventions to save Syracuse from being taken by the Romans.

5. Continue... Archimedes had a good education as a boy, for the Greeks loved knowledge and sent their sons to schools to become knowledgeable Greek citizens. Some of the subjects that he studied as a boy were poetry, politics, astronomy, mathematics, music, art, and military tactics. When Archimedes was in his teens he sailed to Egypt to study in Alexandria. There he went to a famous school of mathematics that had been founded by Euclid. Archimedes studied astronomy, physics, and mathematics with many other great minds of the time. His teacher was named Conon of Samos. Conon was one of the greatest influences that Archimedes had in his life. He taught Archimedes many things about life and science. Archimedes' studies in Alexandria became the foundation on which he built his career as a scientist and mathematician.

6. Death of Archimedes Death of Archimedes 212 BC- When a Roman army under Marcellus attacked the Greek city-state of Syracuse, a seaport on Sicily, Archimedes organized the defense. He invented some mechanical weapons, which he mounted on the city’s walls. These machines were tested in battle. The Romans lost. Marcellus then decided to besiege the city. The Roman army set camp. During a truce, Marcellus studied the city’s walls from the inside. He noticed that one place in the wall did not have traps there. During a celebration, the Syracusans feasted. Marcellus led some Romans to the undefended part of the wall. No one noticed them. Whhen the Romans began looting, the Roman general gave orders that Archimedes was to be spared. Unfortunately, Archimedes was working out a problem. He had not noticed that the city was captured. A Roman soldier came and asked him to go with him. When Archimedes refused to go until he had finished working out his problem, the enraged soldier unsheathed his blade and killed him.

7. Contribution 1. Discovered how to find the volume of a sphere and determined the exact value of pi . 2. Principle of Buoyancy . (It is believe that when he discovered the principle of Buoyancy, he went running through the streets naked shouting ' Eureka' - I have found it ) 3. It is believed that he was actually the first to have invented integral calculus , 2000 years before Newton and Leibniz. 4 . Power of Ten a way of counting that refers to the number of 0's in a number which eliminated the use of the Greek alphabet in the counting system. ( Scientific Notation ) 5. A formula to find the area under a curve , the amount of space that is enclosed by a curve.

8. Continue... Archimedes was a great mathematician of ancient times. His greatest contributions were in geometry. He also spent some time in Egypt, where he invented the machine now called Archimedes' screw, which was a mechanical water pump. Among his most famous works is Measurement of the Circle , where he determined the exact value of pi between the two fractions, 3 10/71 and 3 1/7. He got this information by inscribing and circumscribing a circle with a 96-sided regular polygon. Archimedes made many contributions to geometry in his work on the areas of plane figures and on the areas of area and volumes of curved surfaces. His methods started the idea for calculus which was "invented" 2,000 years later by Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Archimedes proved that the volume of an inscribed sphere is two-thirds the volume of a circumscribed cylinder. He requested that this formula/diagram be inscribed on his tomb. His works (that survived) include: A. Measurement of a Circle B. On the Sphere and Cylinder C. On Spirals D. The Sand Reckoner

9. Continue... The Roman’s highest numeral was a myriad (10,000). Archimedes was not content to use that as the biggest number, so he decided to conduct an experiment using large numbers. The question: How many grains of sand there are in the universe? He made up a system to measure the sand. While solving this problem, Archimedes discovered something called powers. The answer to Archimedes' question was one with 62 zeros after it (1 x 10 62 ) When numbers are multiplied by themselves, they are called powers. Some powers of two are: 1 = 0 power=2 0 2 = 1 st power=2 1 2 x 2 = 2 nd power (squared)=2 2 2 x 2 x 2= 3 rd power (cubed)=2 3 2 x 2 x 2 x 2= 4 th power=2 4 There are short ways to write exponents. For example, a short way to write 81 is 3 4 .This is read as three to the fourth power. On Plane Equilibriums On Floating Bodies This problem was after Archimedes had solved the problem of King Hiero’s gold crown. He experimented with liquids. He discovered density and specific gravity .

10. On the sphere and cylinder In the first book of On the sphere and cylinder Archimedes shows that the surface of a sphere is four times that of a great circle , he finds the area of any segment of a sphere, he shows that the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and that the surface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases. A good discussion of how Archimedes may have been led to some of these results using infinitesimals is given in [14]. In the second book of this work Archimedes' most important result is to show how to cut a given sphere by a plane so that the ratio of the volumes of the two segments has a prescribed ratio.

11. On spirals In On spirals Archimedes defines a spiral, he gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved. He gives results on tangents to the spiral as well as finding the area of portions of the spiral. In the work On conoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse either about its major axis or about its minor axis. The main purpose of the work is to investigate the volume of segments of these three-dimensional figures. Some claim there is a lack of rigour in certain of the results of this work but the interesting discussion in [43] attributes this to a modern day reconstruction. This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant velocity. Equivalently, in polar coordinates(r , θ) it can be described by the equation r = a + bθ with real numbers a and b. This is an early example of a mechanical curve (a curve traced by a moving ) considered by a Greek mathematician.

12. The Sandreckoner The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to 8 × 10 63 in modern notation. He argues in this work that this number is large enough to count the number of grains of sand which could be fitted into the universe. There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could contain. He states that Aristarchus has proposed a system with the sun at the centre and the planets, including the Earth, revolving round it. In quoting results on the dimensions he states results due to Eudoxus , Phidias (his father), and to Aristarchus . There are other sources which mention Archimedes' work on distances to the heavenly bodies. For example in [] Osborne reconstructs and discusses:- ...a theory of the distances of the heavenly bodies ascribed to Archimedes, but the corrupt state of the numerals in the sole surviving manuscript [ due to Hippolytus of Rome, about 220 AD ] means that the material is difficult to handle

13. On plane equilibriums The treatise On plane equilibriums sets out the fundamental principles of mechanics, using the methods of geometry. Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and these are given in this work. In particular he finds, in book 1, the centre of gravity of a parallelogram, a triangle, and a trapezium. Book two is devoted entirely to finding the centre of gravity of a segment of a parabola. In the Quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord. The first book is in fifteen propositions with seven , while the second book is in ten propositions. In this work Archimedes explains the Law of Lever, stating, "Magnitudes are in equilibrium at distances reciprocally proportional to their weights." Archimedes uses the principles derived to calculate the areas and of various geometric figures including triangles, Parallelogram and parabolas. Archimedes is said to have remarked of the : Give me a place to stand on, and I will move the Earth.

14. On floating bodies On floating bodies is a work in which Archimedes lays down the basic principles of hydrostatics. His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes' principle , is contained in this work. He also studied the stability of various floating bodies of different shapes and different specific gravities.

15. Measurement of the Circle In Measurement of the Circle Archimedes shows that the exact value of π lies between the values 3 10 / 71 and 3 1 / 7 . This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides. This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of . In Proposition II, Archimedes shows that the value of (pi) is greater than 223⁄71 and less than 22⁄7. The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when only a rough figure is required.

16. Continue... Archimedes’ Principle: When a body is immersed in liquid, it receives an upward thrust from the bottom towards the top equal to the weight of the displaced liquid. The Sphere and Cylinder The ratio of the volume of a sphere is two-thirds the volume of a circumscribed cylinder (Circumscribed refers to a shape surrounding another shape. Inscribed refers to a shape inside another shape).

17. Pictures:Don't Disturb my Circles

18. Eureka !

19. Give me a place to stand on, and I will move the Earth

20. Heat Ray

22. Euclid

25. Biography Euclid is one of the world's most famous mathematicians, yet very little is known of his life, except that he taught at Ptolemy’s university at Alexandria, Egypt. Euclid's Elements , a work on elementary geometry and other topics, exceeded other works of its time, which are now known only by indirect reference. The Elements begins with definitions, postulates, and axioms, including the famous fifth, or parallel, postulate that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this postulate not demonstrable assumption led to Euclidean geometry. It was not until the 19th century that the fifth postulate was modified in order to develop non-Euclidean geometry

26. Contribution The Elements are divided into 13 books. The first 6 are on geometry; 7, 8 and 9 are on number theory; and book number 10 is on Eudoxus's theory of irrational numbers. Books 11, 12, and 13 all concern solid geometry, and end with a discussion of the properties of the five regular polyhedrons and proof that there can only be these five. Euclid's Elements are remarkable for the clarity with which the theorems and problems are selected and ordered. The propositions proceed logically and concisely, with very few assumptions.

27. Continue... Euclid is not known to have made any original discoveries, and the Elements is based on the work of the people before him, like Exodus, Thales, Hippocrates, and Pythagoras. It is accepted that some of the proofs are his own and that the excellent arrangement is his. Over a thousand editions of the work have been published since the first printed version of 1482. Euclid's other works include Data, On Divisions of Figures, Phaenomena, Optics, Surface Loci, Porisms, Conics, Book of Fallacies, and Elements of Music . Only the first four of these have survived.

28. Life Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.

29. Continue... The few historical references to Euclid were written centuries after he lived, by Proclus and Pappus of Alexandria . [4] Proclus introduces Euclid only briefly in his fifth-century Commentary on the Elements , as the author of Elements , that he was mentioned by Archimedes , and that when King Ptolemy asked if there was an easier path to learning geometry than Elements , Euclid replied, " Sire, there is no Royal Road to Geometry ." [ cite this quote ] Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes. [5][6] In addition, the "royal road" anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Grea t . [7] In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought." [8] It is further believed that Euclid may have studied at Plato's Academy in Athens .

30. Elements One of the oldest surviving fragments of Euclid's Elements , found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5. [9] Main article: Euclid's Elements

31. Continue... Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later. [10] There is no mention of Euclid in the earliest remaining copies of the Elements , and most of the copies say they are "from the edition of " or the "lectures of Theon", [11] while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

32. Continue... Although best-known for its geometric results, the Elements also includes number theory . It considers the connection between perfect numbers and Mersenne primes , the infinitude of prime numbers , Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry , and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

33. Other Works Euclid's construction of a regular dodecahedron Construction of a dodecahedron basing on a cube

34. Continue... In addition to the Elements , at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements , with definitions and proved propositions. Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements . On Divisions of Figures , which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios . It is similar to a third century AD work by Heron of Alexandria . Catoptrics , which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its author may have been Theon of Alexandria . Phaenomena , a treatise on spherical astronomy , survives in Greek; it is quite similar to On the Moving Sphere by , who flourished around 310 BC.

35. Continue... Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. believed these results to be important in astronomy and included Euclid's Optics , along with his Phaenomena , in the Little Astronomy , a compendium of smaller works to be studied before the Syntaxis ( Almagest ) of Claudius Ptolemy .

36. Continue... Other works are credibly attributed to Euclid, but have been lost. Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost. Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial. Pseudaria , or Book of Fallacies , was an elementary text about errors in reasoning . Surface Loci concerned either loc i (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces .

37. Continue... Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

38. Pictures Statue of Euclid in the Oxford University Museum of Natural History

39. Equation To find out the total number of degrees in a polygon with N sides, all you have to do is to use the equation. (N-2) x (180) To find out the degrees in each angle if the polygon is regular (sides are the same length and the degrees in each angle is the same), you just use the formula above and divide by the number of sides the polygon.

40. Quiz 1.) What are the total number of degrees in a rectangle? 90 O 180 O 360 O 2.) How many degrees are there in one angle of an equilateral (regular) triangle? 60 O 180 O 90 O 3.) How many degrees in all are there in a dodecagon (a twelve-sided polygon)? 1980 O 1800 O 2060 O

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