While the study of curvature is an ancient one, the geometry of curved surfaces is a topic that has been slowly developed over centuries. The Ancient Greeks certainly considered the curvature of a circle and a line distinct, noting that lines do not bend, while circles do. Aristotle expanded on this concept explaining that there were three kinds of loci: straight, circular, and mixed (Coolidge)Then in the third century B.C. Apollonius of Perga found that at each point of a conic section there is exactly one normal line (Coolidge, 375-6). However, the Greeks had little to more to offer in the study of curvature.
In the fourteenth century, it was Nicolas Oresme who gave the a definition of curvature. Oresme defines “Curvitas” as follows: if there are two curves touching the same line at the same point then the smaller curve will have greater curvature (Coolidge, 376). After Oresme, there is a gap of nearly 300 years before Kepler began again the discussion of curvature. Kepler discovered that if one takes “a curve all of whose tangents are on one side. To this is attached a flexible string which is pulled taut and then unwound, the curve from which the string springs is called the evolute” (Coolidge, 377). From here, he provides the theorem that states if two curves have tangents on one side which have a common point cannot have the same set of normals (Coolidge (377). While his theorems because quite useful, his ignorance of the calculus did not allow him to extend his theorems to the general cases.
Then Sir Issac Newton in the 17th century worked to develop the concept of the center of curvature which is the center of a circle having the same curvature, and no other tangent circle can lie between this and the c...
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Ramsay, Arlan, and Robert D. Richtmyer. Introduction to hyperbolic geometry. Springer, 1995.
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a.k.a. a.k. Web. The Web. The Web. 09 Oct. 2013.
Einstein’s Special Theory of Relativity has had a colossal impact on the world and is the accepted physical theory reg...
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
In 1543 Nicholas Copernicus, a Polish Canon, published “On the Revolution of the Celestial Orbs”. The popular view is that Copernicus discovered that the earth revolves around the sun. The notion is as old as the ancient Greeks however. This work was entrusted by Copernicus to Osiander, a staunch Protestant who though the book would most likely be condemned and, as a result, the book would be condemned. Osiander therefore wrote a preface to the book, in which heliocentrism was presented only as a theory which would account for the movements of the planets more simply than geocentrism did, one that was not meant to be a definitive description of the heavens--something Copernicus did not intend. The preface was unsigned, and everyone took it to be the author’s. That Copernicus believed the helioocentric theory to be a true description of reality went largely unnoticed. In addition to the preface, this was partly because he still made reassuring use of Ptolemy's cycles and epicycles; he also borrowed from Aristotle the notion that the planets must move in circles because that is the only perfect form of motion.
The first record of the movement of the planets was produced by Nicolaus Copernicus. He proposed that the earth was the center of everything, which the term is called geocentric. Kepler challenged the theory that the sun was the center of the earth and proposed that the sun was the center of everything; this term is referred to as heliocentric. Kepler’s heliocentric theory was accepted by most people and is accepted in today’s society. One of Kepler’s friends was a famous person named Galileo. Galileo is known for improving the design and the magnification of the telescope. With improvement of the telescope Galileo could describe the craters of the moon and the moons of Jupiter. Galileo also created the number for acceleration of all free falling objects as 9.8 meters per second. Galileo’s and Kepler’s theories were not approved by all people. Their theories contradicted verses in the bible, so the protestant church was extremely skeptical of both Galileo and Kepler’s
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1 - Concentric theory - 15th century - taught that sun, planets revolved around the earth.
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The first person in the book was Sir Isaac Newton. Newton was a man that had deep depression and mostly kept to himself. If not for that quality he may not have made the discoveries that he did. He would often sit in the garden for hours on end just thinking and formulating his ideas about the universe. In fact, that is the very place where the ideas of gravity and centrifugal force first came to him. He noticed an apple fall, and wondered why the apple fell to the earth but the moon didn’t. The main discovery that Newton is credited with is the Universal Law of Gravitation. In the prologue, the book describes how this equation told scientists in NASA how to escape gravity and leave the earth to go to the moon. The Universal Law of gravitation is a fundamental law of the world today.
... The Edge of Infinity: Supermassive Black Holes in the Universe. Cambridge: Cambridge UP, 2003. Print.
In his book, Repcheck recounts how a Catholic Church cleric invented a highly complicated theory of the heavens’ architecture. Copernicus made a breakthrough by solving a significant astronomical problem. Everybody except the astronomers had earlier accepted Aristotle’s concept that heavenly objects revolved around the earth in perfectly circular orbits. The astronomers were opposed to this notion since their calculations could not work according to it. Repcheck introduces Ptolemy who described a cosmos in which the earth positioned itself somewhat off-center and other heavenly bodies revolved in one circular orbit inside a second ideal circle at changeable speeds. Even though Ptolemy’s model was rather complicated, astronomers found it to be reasonable in their calculations. Astronomers were still using this new concept even 1500 years later. In this regard, the author starts to bring Copernicus into the picture.
In 1905, Einstein’s Theory of Special Relativity was proposed. The reason that it is so "special" is because it was part of the more complex and extensive Theory of General Relativity, which was published in 1915. His theory reshaped the world of physics when it contradicted all previous laws of motion erected by Galileo and Newton. By mathematically manipulating these previous laws of motion, physicists in the nineteenth century were able to explain such phenomena as the flow of the ocean, the orbits of planets around the sun, the fall of rocks, and the random behavior of molecules in gases. At first, Einstein faced great opposition when he came up with his radical new theory because the previous laws of motion proposed by Galileo and expanded upon by Newton had remained valid for over two hundred years. However, it wouldn’t be long before the "cement" in the foundation of Newtonian and Galilean physics would begin to crumble.
... be separate in what came to be known as synthetic geometry (geometry without algebra, dealing with proofs, axioms, theorems, and postulates). La Geometrie also had the unfortunate fate of being on the list of forbidden works by the church. This was due to Rene Descartes's meditations, which seemingly liberated Europe from Church thinking and thus went against the Catholic teaching. Descartes was excommunicated by the church, and they condemned all of his works, which would slow the spreading of analytical geometry.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
Conic sections are the various gemetric figures created by the interection of a plane. They are among the oldest curves in history and is one of the oldest area of study for mathmaticians. conics were discovered by Menaechmus (c. 375 - 325 BC), a Greek pupil of Plato and Exodus. He was trying to solve the famous problem duplicating a cube. Euclid studied them and Appollonius reinforced and expanded previous results of conics into a book he named Conic Sections. It is a series of eight books with 487 propositions. He applied his findings to the study of planetary motion and it was used to advance the development of Greek astronomy. It is because of Appollonius that the name ellipse, parabola, and hyperbole were given to conics. Conics evolved even further during the Renaissance with Kepler’s law of planetary motion, Descarte on his work Geometry and Fermat’s coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, and Pascal. We can see conics in satellite dishes, sharpening pencils, automobile headlights, when a baseball is hit, telescopes, and much more. Physicians apply conics in treating kidney stones. Even, John Quincy Adams used conics to eaves drop on members of the house of representatives from his desk in the U.S. Capitol building.