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Euclid and the elements essay
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“There is no royal road to geometry.” – Euclid
Euclid’s Elements are predominantly the most fundamental concepts of mathematics, but his perspective on geometry was the model for over two millennia. He is believed by many to be the leading mathematics teacher of all time. However, little is known about his life outside of mathematics, or even when he was born or when he died. According to a passage written by Proclus, Euclid probably lived after Ptolemy and the pupils of Plato, but came before Archimedes and Eratosthenes. This places his existence sometime around 300 B.C. Euclid is most famous for having set the guidelines for geometry and arithmetic written in Euclid’s Elements, a series of thirteen books in which Euclid states definitions, postulates, and theorems for mathematical concepts that are still used today. What is most remarkable about the Elements is the simple, rational, and very logical structure in which Euclid presents the accumulated geometrical knowledge from the past several centuries of Greek mathematicians. The manner in which the propositions have been derived is considered to be the prime model of the axiomatic method. (Hartshorne 296).
Euclid’s axiomatic method works by “starting from a small number of definitions and assumptions at the beginning, [so that] all the succeeding results are proved by logical deduction from what has gone before.” In essence it is no more than “a method of proving that results are correct.” Many of Euclid’s proofs are constructions, all of which can be done using no more than a ruler and a compass and rely only on the theorems and rules of the system. Despite having developed this rigorous system of proofs, Euclid did not actually demonstrate everyt...
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... of Nebraska Press, 1991.
Blumenthal, Leonard M. A Modern View of Geometry. San Fransisco: W.H. Freeman and Company, 1961.
Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries. New York: W.H. Freeman and Company, 1993.
Hartshorne, Robin. Geometry, Euclid and Beyond. New York: Springer, 2000.
Hofstadter, Douglas R.. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1975
Narkiewicz, Wladyslaw. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Brlin: Springer-Verlag, 2000.
Singer, David A. Geometry: Plane and Fancy. New York: Springer, 1998.
Internet Sources:
Joyce, D. E. “Euclid’s Elements.” 1997, [online]. http://aleph0.clarku.edu/~djoyce/java/elements/toc.html (September 18, 2002)
Modeste Petrovich Mussorgsky’s (1839-1881) Songs and Dances of Death was his final composition, composed in 1877, in years of artistic confidence that followed the success of his masterpiece, the opera Boris Godunov. Boris Godunov encapsulates many of Mussorgsky’s innovations including those towards his approach to the setting of the Russian language: his biographer, Robert W. Oldani observes, Mussorgsky’s “quest to find a musical equivalent for the patterns, inflections, pace and cadence of spoken Russian, to fix in music the paralexical aspects of speech that give it plasticity and nuance.” Indeed, Mussorgsky was one of a group of Russian composers known as the "Mighty Five,"- so dubbed by the influential contemporaneous critic V. V. Stasov.
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.
- Norris, Jeremy Paul. The development of the Russian piano concerto in the nineteenth century. Indiana: Indiana University Press, 1988. Print.
One source of this interest in method was ancient mathematics. The thirteen books of Euclid's Elements was a model of knowledge and deductive method. But how had all this been achieved? Archimedes had made many remarkable discoveries. How had he come to make these discoveries? The method in which the results were pr...
Fiero, Gloria K. “Greece: Humanism and the Spculative Leap” The Humanistic Tradition: The First Civilizations and the Classical Legacy. McGraw Hill. 6th ed. New York: New York, 2011. 76-134. Print.
The music industry is about selling the recordings and the performances of music among many individuals and organizations that works with musicians. The music industry has both the charm and the organizational architecture to affect the business and the cultural trends. There are millions of listeners and followers to the products of the music industry. The opinions of these listeners always change depending on what they see or hear. The music business or industries have been influencing generations of recording artists, business professionals and consumers. The music industries have been reinforcing questionable subject matter, music lyrics and business and social norms.
Robert, A. Wayne and Dale E. Varberg. Faces of Mathematics. New York: Harper & Row Publishers, Inc., 1978.
The production and consumption of music has become a market of its own that has been growing and getting stronger every single day. Many companies have found their fortune in the music industry and have become part of the globalization by entering the worldwide market and economy.
...t funeral/death. When cut the nail at night, you wouldn't be with your parent when they are dying. If you are sleep toward the north, then your bodies are laid down. Japanese do have more of these superstitions.
This source provided a lot of background information on Euclid and his discoveries. This source gave details about the many geometrical theories of Euclid, as well as his practical geometrical uses. This source also explained how geometry helped Greece a long time ago, and how it is used by many people everyday.
Abstract: Relatively little attention has been given to the effect of digital music on amateur musicians and music distribution. Here, I examine the revolution on the horizon-sites such as MP3.com herald the eventual bridging of the gap between artist and listener while shrinking the record companies. In this paper I examine two such sites that host independent and labeled artists alike: the larger, better-known MP3.com and the smaller, independent, non-profit Songfight. I examine how they each handle the challenges of digital music and their attraction for artists while concluding that a change in the recording industry as we know it is forthcoming.
Euclid of Alexandria was born in about 325 BC. He is the most prominent mathematician of antiquity best known for his dissertation on mathematics. He was able to create “The Elements” which included the composition of many other famous mathematicians together. He began exploring math because he felt that he needed to compile certain things and fix certain postulates and theorems. His book included, many of Eudoxus’ theorems, he perfected many of Theaetetus's theorems also. Much of Euclid’s background is very vague and unknown. It is unreliable to say whether some things about him are true, there are two types of extra information stated that scientists do not know whether they are true or not. The first one is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre. This is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors. The next type of information is that Euclid was born at Megara. But this is not the same Euclid that authors thought. In fact, there was a Euclid of Megara, who was a philosopher who lived approximately 100 years before Euclid of Alexandria.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
Euclid, also known as Euclid of Alexandria, lived from 323-283 BC. He was a famous Greek mathematician, often referred to as the ‘Father of Geometry”. The dates of his existence were so long ago that the date and place of Euclid’s birth and the date and circumstances of his death are unknown, and only is roughly estimated in proximity to figures mentioned in references around the world. Alexandria was a broad teacher that taught lessons across the world. He taught at Alexandria in Egypt. Euclid’s most well-known work is his treatise on geometry: The Elements. His Elements is one of the most influential works in the history of mathematics, serving as the source textbook for teaching mathematics on different grade levels. His geometry work was used especially from the time of publication until the late 19th and early 20th century Euclid reasoned the principles of what is now called Euclidean geometry, which came from a small set of axioms on the Elements. Euclid was also famous for writing books using the topic on perspective, conic sections, spherical geometry, number theory, and rigor.