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GREEK MATHEMATICS Greek was very clumsy in writing down the numbers. They didn’t like algebra. They found it very hard to write down equations or number problems. Instead Greek mathematicians were focused on geometry, and used geometric methods to solve problems that you might use algebra for. Greek mathematicians were very interested in proving that certain mathematical ideas were true. They spent a lot of time using geometry to prove that things were always true,even thoughpeople like Egyptians and Babylonians already knew that they were true most of the time away. The Greeks in general were very interesed in rationality, in things making sense and handing together. Music were very important for them, because it followed strict rules to produce beauty. Some famous Greek mathematicians were: Pythagoras, Aristotle, Anaxagoras, Thales, Antiphon, Archytas, Democritus, Euclid, Hipocrates, Plato, Xenocrates, Zeno,Socrates... Early Greek Apppreciaton of Geometric Forms -use of crude parallels -less crude and more elaborate forms -more delicate forms Greek Algebra The Greeks proved that (a+b) ² = a²+2ab+b² They had no algebraic shorthand and consider only lines and rectangles instead of numbers and products. They knew such other identities as (a+b) (a-b) = a²-b² a(x+y+z) = ax+ay+az (a-b) ² = a²-2ab+b² They could complete the square of binomial expression a²±2ab Origin of Greek Mathematics -three important periods in the development of Greek mathematics The periods may be characterized as: First - the one subject to the inluence of Pythagoras Second - the one dominated by Plato and his school Third- the one in which the Alexandrian School flourished in Grecian Egypt and extended its influence to Sicily, the Ǽgean Islands and Palestine Thales -the first of the Greeks who took any scientific interest in mathematics in general -merchant, statesman, mathematician, astronomer, philopher Arithmetic of Thales -he knew many number relations -in his work is founding deductive geometry Geometry of Thales -he is credited with a few of the simplest propositions relating to the plane figures 1. Any circle is bisected by its diameter. 2.The angles at the base of an isosceles triangle are equal. 3. When two lines intersect, the vertical angles are equal. 4. An angle in a semicircle is a right angle. 5. The sides of similar triangles are proportional. 6. Two triangles are congruent if they have two angles and a side respectively equal. -his great contribution lay in suggesting a geometry of lines and in making the subject abstract -he gave the idea of a logical proof as applied to geometry Anaximander -the leadership of the Jonian School -he brought the gnomon in Greece, and used for determining noon Pythagoras -he had been Thales pupil -the familiar proposition in geometry that bears his name was known, as already started, in India, China and Egypt -he had two groups of the disciples: the hearers and the methematicians -he asserted that unity is the essence of number Geometry of Pythagoras -he investigated his theorems from the immaterial and intellectual point of view -he discovered the theory of irational quantities and the construction of the mundane figures -he defined a point as unity having position Zeno It is important Zeno’s paradoks. It talks about that Achilles cuold not pass a tortoise, even thogh he went faster than tortoise. Agatharchus He showed how to make use of notion of projection upon a plane surface. Socrates He should be mentioned in connection with the early development of a logical geometry. Enopides He discovered two problems of Euclid, one referring to the drawing of a perpendicular to a given line from an external point,and the other referring to the making of an angle equal to a given anlge. Democritus He was the first to show the relation between the volume of a cone and that of a cylinder of equal base and equal height, and similarly for the pyramid and prism. Hippias of Elis He invented a simple device for trisecting any angle, this device being known as the quadratrix. Hippocrates He arranged the propositions of geometry in a scientific fashion. He discovered the first case of quadrature of a curvilinear figure, namely, the proof that the sum of the two shaded lines here shown is equal to the shaded triangle. The Method of Exhaustion -the area between a curvilinear figure (e.g. a circle) and a rectilinear figure (e.g. an inscribed regular polygon) could be aproximately exhausted by increasing the number of sides of the latter Antiphon He inscribed a regular polygon in a circle,doubled the number of sides, and continued doubling until the sides finally coincided with the circle. Since he could construct a square equivalent to any polygon, he could then construct a square equivalent to the circle; that is, he could “square the circle”. Archytas 1. If a perpendicular is drawn to the hypotenuse from the vertex of the right angle of a right angled triangle, each side is the mean proportional betwen the hipotenuse and its adjecent segment. 2. The perpendicular is the mean proportional between the segments of the hypotenuse. 3. If the perpendicular from the vertex of a triangle is the mean proportional between the segments of the opposite side, the angle at the vertex is a right angle. 4. If two chords intersect, the rectangle of the segments of one is equivalent to the rectangle of the segments of the other. 5. Angles in the same segment of a circle are equal. 6. If two planes are perpendicular to a third plane their line of intersection is perpendicular to that plane and also to their lines of intersection with that plane. Theæte’tus He discovered a cosiderable part of elementary geometry and wrote upon solids. Plato -the method of analysis -interested in arithmetic -mystcism of numbers -60 - Platonic number -accurate definitions, clear assumptions, logical proof Speusip’pus -wrote upon Pythagorean numbers -wrote upon proportion -rare elegance the subjects of linear, polygonal, plane, and solid numbers Xenoc’rates -deified unity and duality -assumed the existence of indivisible lines Ar’istotle He wrote two works of a mathematical nature. Continuity: “A thing is continuous when of any two successive parts the limits at which they touch are one and the same and are, as the word implies, held together.”