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Greek_Mathematics_II - Greek Mathematics Greek Mathematics...

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Unformatted text preview: Greek Mathematics Greek Mathematics The Period from Thales to Euclid 600 B.C. – 300 B.C. Lines of Mathematical Lines of Mathematical Development • Three important and distinct lines of development during the first 300 years of Greek mathematics. 1. Development of material that was organized into the Elements. 2. Development of higher geometry, the geometry of curves and surfaces other than circles, spheres and planes. Development of Mathematics Development of Mathematics 1. Development of notions connected with infinitesimals and with limit and summation processes that did not attain final clarification until after the invention of calculus. Paradoxes of Zeno Method of exhaustion of Eudoxus • • Quadrature of a Plane Figure Quadrature of a Plane Figure • The Pythagoreans were interested in transforming an area from one shape into another rectilinear shape. • For all their geometric constructions, the Greeks used a compass and an unmarked straightedge (ruler). • These two tools help the geometer construct a straight line and a circle. Quadrature Quadrature • The quadrature or squaring of a plane figure is the construction – using only compass and straightedge – of a square having area equal to that of the original plane figure. • If the quadrature of the plane figure is possible, we say that the figure is quadrable (or squarable). Quadrature Quadrature • Quadrature of a rectangle – construct a square with the same area as a given rectangle. (possible) • Quadrature of a triangle – construct a square with the same area as a given triangle. • Quadrature of a polygon – irregular polygons are quadrable. Duplication, Trisection and Duplication, Trisection and Quadrature • Besides the Ionian school founded by Thales and the early Pythagorean school at Crotona, other mathematical centres arose and flourished in Asia Minor and Italy. • After the Persian wars, Athens became the centre for intellectual development. Many were attracted from all parts of the Greek world. Greek Mathematicians Greek Mathematicians • Anaxagoras, the last member of the Ionian school settled in Athens. Many of the dispersed Pythagoreans found their way to Athens and Zeno and Parmenidis of the Eleatic school went to Athens to teach. • Hippocrates from the island of Chios visited Athens.(Hippocrates of Cos became the father of Greek medicine and originator of the Hippocratic oath) Hippocrates of Chios Hippocrates of Chios • Remembered for two contributions to geometry: 1. The development of theorems of geometry precisely and logically from a few given axioms and postulates. 2. Quadrature of the lune. • A lune is a plane figure bounded by two circular arcs. Quadrature of the lune Quadrature of the lune • Hippocrates did not square all such 1. 2. 3. figures but rather a particular lune he had carefully constructed. His proof rested upon the following three results: The Pythagorean Theorem. An angle inscribed in a semicircle is right The areas of two circles or semicircles are to each other as the squares on their diameters. Plato 427 – 347 B.C. Plato 427 – 347 B.C. • Plato was born in Athens. He studied philosophy under Socrates. • Primarily a philosopher – traveled the world formulating his ideas. • He studied mathematics under Theodorus of Cyrene in Africa and became a good friend of Archytas (a gifted Pythagorean) • Retuned to Athens and founded the Academy (modern day university). Plato Plato • Plato was president of the Academy for the rest of his life. • The Academy attracted scholars from all around the world. Under his guidance, became an intellectual centre. • Almost all of the important mathematical work of the 4th century B.C. was done by friends and students of Plato. The Academy The Academy • The Academy became the link between mathematics of the earlier Pythagoreans and that of the later school of Mathematics at Alexandria. Many subjects were studied at the Academy but mathematics was at the top. Plato’s influence on Mathematics was not because of any discoveries he made but rather to his strong conviction of its importance. • • The Academy The Academy • According to Plato, the study of mathematics furnished the finest training for the mind and hence was essential for the cultivation of philosophers. He regarded geometry as the ideal entrance requirement to the Academy. Across the arched entry to the Academy were the words ”Let no man ignorant of geometry enter here”. • The Republic The Republic • The mathematical syllabus that Plato developed for students at the Academy is described in his most famous work, The Republic. Syllabus consisted of five subjects, arithmetic, plane geometry, solid geometry, astronomy and harmonics (music). • Academy produced many capable mathematicians including Eudoxus . Eudoxus of Cnidos Eudoxus of Cnidos • He was a student of Plato and Archytas. He founded a school at Cyzicus in Northern Asia Minor. • Eudoxus was interested in Astronomy. He gave complex explanations of lunar and planetary motion whose influence was felt until the Copernicus revolution in the 16th century. Eudoxus’ contributions Eudoxus’ contributions • Eudoxus is remembered for two major contributions in mathematics: 1. The theory of proportion. 2. The method of exhaustion. • After Hippasus proved that not all line segments are commensurable, many proofs that the Pythagoreans developed were not correct. Theorems were correct but proofs were incorrect. Theory of Proportion Theory of Proportion • Eudoxus developed a valid theory of proportions and supplied the proofs to those theorems. His theory can be found in Book V of Euclid elements. His theory of proportion says if A, B, C, D are any four unsigned magnitudes, A and B being of the same kind and C and D being of the same kind, then the ratio of A to B is equal to that of C to D when for arbitrary positive integers m and n, mA = nB < > • as mC = nD < > Theory of Proportion Theory of Proportion • The Eudoxian theory of proportion provided a foundation, later developed by Dedekind and Weierstrass, for the real number system of mathematical analysis. • The method of exhaustion had an immediate application in the determination of areas and volumes of more complex geometric figures. Method of Exhaustion Method of Exhaustion • The general strategy was to approach an irregular figure by means of succession of known elementary ones, each providing a better approximation. • For example, to approximate the area of a circle, inscribe within it a square, then double the number of sides of the square to get an octagon, then double again to get a 16­ogon and so on. Method of Exhaustion Method of Exhaustion • These simple polygons can closely approximate the circle. • In Eudoxus terms, the polygons are “exhausting the circle from within”. • The modern notion of “limits” is based on the method of exhaustion. Other Mathematicians of the Other Mathematicians of the Academy • Menaechmus – student of Eudoxus, invented the conic sections. • Dinostratus – brother of Menaechmus, an able geometer and student of Plato. • Theaetetus – a man of unusual gifts to whom we owe much of the materials of Euclid’s book 10 and 13. Aristotle 384 – 322 B.C. Aristotle 384 – 322 B.C. • Aristotle studied at Plato’s Academy and remained there until Plato’s death in 347 B.C. He was then invited to the palace of Philip II of Macedonia to educate his son Alexander. • Then he returned to Athens where he founded his own school, the Lyceum and spent the rest of his days lecturing and writing. Aristotle Aristotle • Aristotle wrote on many subjects, politics, ethics, epistemology, physics and biology. As for mathematics, his strongest influence was in the area of logic. • Great thinkers even before Aristotle were developing the notion of logical reasoning. • Various detailed techniques of argument were developed. Logical Reasoning Logical Reasoning • Examples of such techniques are: • Reductio ad absurdum – one assumes that a proposition to be proved is false and then derives a contradiction. • Modus tollens – one shows first that if A is true, then B follows, shows next that if B is not true, then concludes that A is not true. The Three Famous Problems The Three Famous Problems • 1. 2. 3. There are three problems that originated from the Greek mathematicians and have not been solved even today. They are: The duplication of the cube – the problem of constructing a cube having twice the volume of a given cube. The trisection of an angle – the problem of dividing a given arbitrary angle into three equal parts. The quadrature of a circle – construct a square having an area equal to that of a circle. Euclidean Tools Euclidean Tools • These problems cannot be solved with a straightedge and compass. These tools are known as the Euclidean Tools. • The energetic search for solutions to these problems led to many discoveries such as conic sections, many cubic and quartic curves, algebraic numbers and group theory. Euclidean Tools Euclidean Tools • “With the straightedge, we are permitted to draw a straight line of indefinite length through any two given distinct points. With the compass, we are permitted to draw a circle with any given point as center and passing through any given second point”. • We consider the three problems separately. Duplication of the Cube Duplication of the Cube • The mythical king Minos was dissatisfied with the size of a tomb build for his son. Minos ordered that the tomb be doubled in size. An ancient Greek poet suggested incorrectly that this can be accomplished by doubling each dimension of the tomb. • This faulty mathematics led geometers to study the problem of doubling a given solid and keeping the same shape. Duplication of the Cube Duplication of the Cube • This problem was studied at Plato’s Academy • and there are solutions attributed to Eudoxus and Plato himself. The first real progress was the reduction of the problem by Hippocrates to the construction of two mean proportionals between two given line segments of lengths s and 2s. Duplication of the Cube Duplication of the Cube • If we denote the two mean proportionals by x and y, then s x y == , x y 2s ⇒ x 2 =sy , y 2 =2 sx • x3 = s3 2 From these, we have , thus x is the edge of a cube having twice the volume of the cube on edge s. Duplication of Cube Duplication of Cube • After Hippocrates, one of the remarkable • • • solution of this problem was given by Archytas (400 B.C.). The solution of Eudoxus (370 B.C.) is lost. Menaechmus (350 B.C.) gave two solutions and in the process invented the conic sections. Eratosthenes(230 B.C.) and Nicomedes gave two other solutions. Trisection of an Angle Trisection of an Angle • This problem is the simplest of all three and since the bisection of an angle is so simple, it is natural to wonder why trisection is not equally easy. • The trisection problem was reduced by the Greeks to what they called a verging problem. Trisection of an Angle Trisection of an Angle • Various curves have been discovered that will solve the verging problem. One of the oldest is the conchoid invented by Nicomedes (240 B.C.) • There are other curves that will trisect an angle such as the spiral of Archimedes. Quadrature of a Circle Quadrature of a Circle • The problem of constructing a square with area equal to a given circle goes as far back as 1800 B.C. • Anaxagoras (499 – 427 B.C.) was the first to be connected to this problem. Contribution unknown. • Hippocrates succeeded in squaring some lunes and claimed he could square the circle. Quadrature of a Circle Quadrature of a Circle • After Hippocrates, Hippias of Elis invented the • • • curve known as the quadratix. This curve solves both the trisection and the quadrature problem. A solution of the quadrature problem can also be achieved with the spiral of Archimedes. Since then, countless solutions were proposed. At the end, each was found to contain an error. Mathematicians began to suspect that the quadrature of a circle was impossible. Quadrature of a Circle Quadrature of a Circle • The crucial issue here is not whether such a square exists, but whether it can be constructed with compass and ruler. • This problem remained unsolved until 1882, when the German mathematician Ferdinand Lindemann(1852 – 1939) succeeded in proving that the quadrature of the circle was impossible. Algebraic and Transcendental Algebraic and Transcendental numbers • A real number is called algebraic if it is the solution of some polynomial equation. • A real number is called transcendental if it is not the solution of any polynomial. • A subset of the algebraic numbers can be constructed using Euclidean tools. • Transcendental numbers cannot be constructed using a compass and ruler. ...
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