Greek Mathematics zDevelopment of material later organized into the Elements zNotions of infinitesimals,limits,summation process zGeometry of curves other than the circle and straight line zGeometry originated from attempts to solve three famous problems
Three Famous Problems zDuplication of the cube. The problem of constructing the edge of a cube having twice the volume of a given cube. zTrisection of an angle. The problem of dividing a given arbitrary angle into three equal parts. zThe quadrature of the circle. The problem of constructing a square equal in area to that of a given circle.
Euclidean Tools zStraightedge - draw a straight line of indefinite length through any two points. zCompass - draw a circle with any given point as center and passing through any given second point.
Three Famous Problems zThese problems cannot be solved, except by approximation, using Euclidean Tools. zSearch for solutions led to many mathematical discoveries; conic sections,cubic and quartic curves. zImpossibility was not established until 19th century, more than 2000 years after problems were conceived. zEfforts still continue.
Duplication of the cube zMythical King dissatisfied with the size of a tomb erected for his son. zConsider duplication of a square: yTo double a square with side of length a, we must construct a square with side of length x with twice the area of the given square. yX is the length of the diagonal of the given square.
Duplication of the cube zGiven a cube with edge a,we must construct a cube with edge x with twice the volume of the given cube. zHippocrates reduced this problem to constructing two mean proportionals between line segments of lengths a and 2a.
Trisection of an angle zMost popular among “mathematicians” today zSeveral Greek mathematicians used verging solutions zSolution of Archimedes
Quadrature of the circle zKnown constructions yConstruct a square equal in area to the area of the sum of two given squares yConstruct a rectangle equal in area to a given right triangle yConstruct a right triangle equal in area to a given triangle yConstruct a square equal in area to a given rectangle
Quadrature of the circle zHippocrates of Chios- Quadarture of lunes yLunes - plane regions bounded by the arcs of two different circles yContributed to duplication of cube yFirst textbook in geometry ySquaring lunes helped in attempts to square a circle
Chronology of Pi zComputation of Pi closely tied to quadrature problem zPi - ratio of the circumference of a circle to its diameter zPi is irrational