Squaring the Circle and Trisecting an Angle Essay

Squaring the Circle and Trisecting an Angle - Essay Example Babylonian mathematicians has found the way of constructing a square of equivalent area with respect to given circle. The Indian and Egyptians has also shown method of evaluating which appears in Sulba Sutras and Rhind papyrus of 1800 BC respectively. The Egyptian has calculated the value of approximated to 256/81. Archimedes has better precision which approximates the to be in the range of 3+1/7 and 3+10/71. Anaxagoras was supposed to be a first Greek mathematician who has worked in finding the solution of squaring the circle in the first century of AD. Hippocrates of Chios put an effort to solve it by squaring lunes. Antiphon suggested the method of inscribed polygons in circle and increasing the numbers of sides so that square and circle will have infinitesimally small difference in their area. Eudemus was not convinced with the approach. Bryson augmented the theory of Antiphon by circumscribing the polygon to support its validity. This problem was even discussed in the Birds, play written by Aristophanes around 414 BC. Oenopides, Antiphon, Bryson, Hippocrates, and Hippias were the prominent mathematicians who worked the problem during this period. Oenopides was pioneer in searching for use of compass and ruler as method of constructing a proof. He has contributed the method to draw a perpendicular line. James Gregory tried to prove its impossible nature of construction in 1667. Ferdinand von Lindemann finally succeeded in providing substantial proof in 1882. Hippias and Dinostratus have used the quadratrix to design the method of squaring the circle. This curve is created by mechanical method which produces uniform motion of a line equivalent to rotating radius of a circle in same time. The Greek mathematician has invested enough energy and effort for developing the algorithm of dividing arbitrary angle into desired ratio which will be helpful in constructing regular polygon of n sides. Gauss invented the construction of polygons through ruler and compass although Greeks didn't succeed in achievement of such a mathematical feat. Hippocrates was also aware of method to trisect an angle. Archimedes has significant contribution in many of the mathematical principles mentioned in the Book of Lemmas. His work 'On spirals' has general recognition in on the results provided for trisecting an angle. The contemporary mathematician Nicomedes introduced the concept of conchoids curve to formalize the proof of trisecting an angle. It uses rulers and compass methods where ruler rotates around the conchoids curve with axis fixed on root point of the given line. Another mathematician Pappus referred the solution of trisecting an angle by Apollonius applying the construction of conic. Pappus utilizes the geometry of hyperbola for two of his solutions. Mathematics Applied in Solution of the Geometrical Construction Archimedes applies the construction of spiral curves as method of measurement of the circle. In above drawing Archimedes proved that circumference of the circle with radius OP where