The Life And Work Of Euclid

The Life And Work Of Euclid While studying geometry with Euclid a youth inquired after having learned the first proposition, What do I get by learning these things? Euclid called a slave to them and said, Give him threepence, since he must make a gain out of what he learns. [8] Euclid, a Greek mathematician and teacher, changed the course of the world. Euclids work not only affected the work of other prominent scientists to come after him, but also the lives of ordinary people, which contributed to the rise of modern science in western civilization. What is perplexing is that despite him changing the course of world, we know very little about him. Unlike some other well-known historical figures, Euclids influence did not spread simply by fame. Historians dont even know his exact date of birth. To this day, we do not know which continent he was born on, much less the city. Of the little we do know about Euclid, we know that he taught in Alexandria around 300 B.C. [9], and that he wrote, amongst approximately 10 other books, arguably one of the greatest mathematical textbooks in history, The Elements. The Elements is a geometry textbook that unified all of the previously known principles of geometry. It was unique in that it was constructive in its delivery of its principles. Basically, it explained mathematic principles from the ground up and added onto what was already established. Imagine trying to study science if one concept didnt flow into the next and everything was garbled and out of order. The Elements solved this problem through careful organization and logical delivery of its principles. The Elements wasnt a revolutionary observation or a new and exciting revelation, but rather a book of brilliant deductive reasoning, analysis, and organization. The Elements was explained so well that every Geometry textbook preceding it was practically discarded, and because of this the term Euclidean wasnt necessary or used for over two thousand years because there was no other known form of geometry[17]. Concerning Euclids deductive reasoning and analysis, his axiomatic systems are most prominent. His axiomatic systems are considered to be constructive. [18] This means that he never reached any conclusions or spoke about concepts that he did not yet explain to the reader. He arranged the geometric theorems so that they flowed logically from one to the next. [9] For example, he started with the simplest of concepts such as describing a geometric point and worked his way into derived propositions. [16] More specifically he took a small number of axioms (self-evident logical truths) and deduced many other theorems from them. He even filled in the blanks whenever it was necessary by filling in the missing steps absent from others processes, and even by developing his own proofs [9]. For example, Euclid proved that it is impossible to find the largest prime number. He proved that if you were to take the largest known prime number and 1 to the product of all the prime numbers leading up to it and including it then you will get another prime number. This is accepted as being one of the classics proofs in mathematics because of how clear and concise it is. [5] Euclid put a lot of effort into making it possible for common people to understand geometry rather than just professional mathematicians. How the natural flow and style of explanation of The Elements affected the world is self-evident. Since it is easier to understand scientific concepts when they are communicated clearly and concisely and delivered in a logical order, Euclids book made it much easier for the people to acquire a complete understanding of geometry. As newborns in this world often one of the first things we get to play with are blocks of different geometric shapes. This helps us to develop our minds both visually and mathematically. Euclidean shapes are quite literally everywhere in our society. Unlike Calculus where there is usually a fixed method for solving a given problem, when it comes to geometry, using Euclidean axioms allows people to solve any one problem in several different ways. It also inspires development of problem solving skills. One of the ways Euclidean geometry has been applied and influences our day to day lives is through construction and architecture. For example, if somebody wants to construct a wooden table. If they wanted to figure out if it was square or not they could measure each corner of the table to see if it was at a 90 angle. With Euclidean Geometry, however, they would need only to measure two of the corners. The properties of right triangles within The Elements tells us that if two corners are square then the whole shape is square. This is probably very obvious to a person of our modern day, but it was not at the tme. Unless you are a mathematician you may not even know who such properties can be attributed to and just consider them common knowledge. Another, less obvious way they could have done this is to have measured the distance between two diagonal corners of the table. If the two distances are the same then the table must be a square. The latter method I have described is a common wa y for construction workers or home-improvement workers to check their work. There are countless examples of this that common people can utilize in their everyday lives with the principles of Euclidean Geometry. Euclids influence doesnt end there. Examples of Euclidean geometry can be found in modern day computer graphics. It is used in mesh generation. A mesh is basically a combination of geometric polygons or polyhedrons that create the illusion of a curve. Although the Euclidean Geometry may be widespread within western civilization, in some third world countries there are houses are constructed as lop-sided indeterminate shapes. This is a real-life example of what our architecture would have looked like without Euclids influence.[4] It is fair to say that the study of Euclids book was one of the main contributing factors to the Scientific Revolution and subsequently the rise of science in Europe rather than in Asia. The Elements made the concept of one principle being built upon another glaringly obvious and, over the course of time, it became considered common knowledge in western civilization. Of course, scientists such as Newton, Copernicus, Kepler, and Galileo played significant roles as well [9], but as Sir Isaac Newton said If I have seen further it is by standing on the shoulders of giants [21]. Euclids book provided for us, not just a shoulder, but an entire foundation built of giants shoulders that would have otherwise been scattered and disorganized. This solid base of knowledge allowed western civilization to reach new heights. For example, when it came to Isaac Newton and his book, Principles Of Natural Philosophy, many of his proofs were set in a geometric form similar to those found in The Elements . [12] As it is with any great work of science, The Elements allows others to build upon it or advance into new areas of discovery. Some men, such as Girolamo Saccehri, have tried to disprove or find flaws in Euclids axioms. Saccehri was an Italian mathematician who in 1733 almost discovered a form of non-Euclidean geometry. He studied for years to find a flaw in Euclids work. He was supposedly on the verge of a breakthrough but gave up before his work came to fruition. It wasnt until about a hundred years later in 1899 that a German mathematician by the name of David Hilbert found another set of geometric axioms that differed from Euclid. [13] Non-Euclidean geometry allows us to describe physical space in new ways. Following Hilbert came another German, by the name of Albert Einstein. Einstein recalls receiving two gifts that had particular influence on him as a child, one a magnetic compass, and the other Euclids The Elements. He referred to The Elements as the holy little geometry book. [3] Another example of a great scientist that was influenced by Euclid is Galileo Galilei. In his old age Galileo told his biographers that while attending the University of Pisa he would nose-drop in on lectures being given by Ostilio Ricci to the court pages on Euclid. These lectures were only available to members of the court so he would try to stay quiet whenever he attended them. His interest in Euclid got the better of him after a while and he approached Ricci to ask him questions on Euclid. Ricci noticed Galileos talent for math and eventually became his teacher. Although Galileo was supposed to be going to college to study medicine, (Galen) he secretly studied mathematics (Euclid) instead. Galileo later used Euclids Book Five, Definition Five, to show how bodies of certain arbitrary weight have weights directly proportional to their volumes. [2] This is one of the best examples how influential Euclids work was to anybody with a mind for mathematics and how he changed the course o f history by capturing the interest of a man such as Galileo. Euclids work also influenced philosophers such as Benedict Spinoza. Benedict Spinoza was a prominent philosopher of 17th century. He wrote the ambitious philosophicical book Ethics where he attempts to provide us with a coherent view of the universe and our place in it. To explain such concepts he used Euclids style of delivery complete with axioms and propositions. Speaking of the system within his book and the style in which he chose to present it in Spinoza said that it was demonstrated in geometrical order. [23] Usually philosophical books were written differently, such as Rene Descartes Meditations that was written like a diary. When it comes to mathematicians I think every mathematician alive since the time of Euclid had to have been influenced by his work in some form or another, but, of some of the most prominent mathematicians, Euclid specifically influenced the work of Bertrand Rusell, Alred North Whitehead, Blaise Pascal, Marin Mersenne , and Adrien-Marie Legendre. Interestingly enough Bertrand Russell, an English 20th century mathematician and logician, used Euclids work to push mathematics into the next level by explaining to people in his book An Essay On The Foundations Of Geometry [11] how Euclidean Geometry was being replaced by more advanced forms of geometry. Both Russell and Whitehead were co-authored the epoch Principia Mathmatica in which they referenced Euclid a number of times as evidence in their work. Pascal, a 17th century French mathematician, received a copy of Euclids Elements as a boy and before the age of 13 he had proven the 32nd proposition of Euclid and discovered a flaw in Rene Descartes geometry [25]. Mersenne, also a 17th century French mathematician, used Euclids proof on prime numbers to develop his own ways or forms as they are called, making it even easier to find large prime numbers. Prime numbers are important to modern day society because they are used in cryptographic software security systems. Basically, large prime numbers can be implemented into coding schemes that are difficult to break. Legendre, a 19th century French mathematician, wrote his most famous book Elà ©ments de Gà ©omà ©trie based entirely off of The Elements. In it he sought to simplify Euclids propositions even further. Elà ©ments de Gà ©omà ©trie was used in elementary school classrooms for over a 100 years. [13][24][6] Euclid influenced politicians such as Abraham Lincoln. Lincoln, as a lawyer traveling on horseback would carry a copy of Euclids Elements in his saddlebag. According to his law partner, at night Lincoln would lay on the floor for hours at night studying Euclids Elements by lamplight. [5] He was a great admirer of the logical delivery of information that The Elements contained and used Euclids systematic approach in many of his speeches. It is no coincidence that the phrase dedicated to the proposition bears such striking similarities to Euclids axioms. Lincoln, speaking of his study of Euclid, said, In the course of my law reading I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? I consulted Websters Dictionary. They told of certain proof, proof beyond the possibility of doubt; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my fathers house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies. [1][5] The astronomers Johannes Kepler and Nicolaus Copernicus were also influenced by Euclids work. When it came to Keplers approach to astronomy he depended almost entirely on Euclid. Kepler, much like Galileo studied Euclid while attending a university (Tà ¼bingen). Kepler was a devout Lutheran and considered Euclid geometry to be the only geometry that could be applied to the heavens and refused to use any other form of geometry because he considered such forms to be heretical. He developed a proof of concerning planetary motion based entirely off propositions found in The Elements [22]. Copernicus used Euclids work on optics as evidence in his book On The Revolutions Of The Celestial Spheres which was considered the starting point of modern astronomy and the defining epiphany that began the scientific revolution. All these great men of science were not able to use Euclids work as evidence simply because he was well known or famous for doing something exciting and spectacular. It was the intellectual quality of Euclids work that made the difference. We dont know enough about Euclid to either love him nor hate him and unless you happen to be a mathematician, his work is undoubtedly not awe inspiring. Nevertheless, Euclid still managed to affect some of the most important figures of the Scientific Revolution by setting the foundations necessary for the development of modern science. Sources: 1. The Lincoln year book, written by Abraham Lincoln, 1809-1865, passage 32 2. Galileo at Work: His Scientific Biography, written by Stillman Drake, pages 2-3 3. Einstein as a Student, written by Dudley Herschbach, page 3 4. How To Use Euclidean Geometry, written by Henri Bauholz, http://www.ehow.com/how_4461018_use-euclidean-geometry.html 5. Euclid, Math Open Reference, http://www.mathopenref.com/euclid.html 6. Great Scientists: from Euclid to Stephen Hawking, written by John Farndon, 2007 7. A Chronicle of Mathematical People, written by Robert A. Nowlan 8. Geometry Quotes, History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/~john/MT4521/Lectures/Q1.html 9. The 100: A Ranking Of The Most Influential Persons In History, written by Michael H. Hart, 2000 10. Encyclopedia of World Biography. Euclid 11. The Teaching of Euclid, written by Bertrand Russell, http://www-history.mcs.st-and.ac.uk/Extras/Russell_Euclid.html 12. Isaac Newton, Wikipedia, http://en.wikipedia.org/wiki/Isaac_Newton 13. Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, written by Luetta Reimer Wilbert Reimer, 1990 14. The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450, written by David C. Lindberg, 2008 15. Mathematics: From the Birth of Numbers, written by Jan Gulberg, 1996 16. Euclids Elements, written by D.E. Joyce, http://aleph0.clarku.edu/~djoyce/java/elements/elements.html 17. Euclid, Wikipedia, http://en.wikipedia.org/wiki/Euclid 18. Axiomatic Systems for Geometry, written by George Francis, 2002 19. The Thirteen Books of the Elements, written by Euclid / Thomas L. Heath 20. Mathmatical Thought, University of Arkansas, http://math2033.uark.edu/wiki/index.php/EuclidHYPERLINK http://math2033.uark.edu/wiki/index.php/Euclids_ElementsHYPERLINK http://math2033.uark.edu/wiki/index.php/Euclids_Elementss_Elements 21. Newton: Understanding the Cosmos, New Horizons, Letter from Isaac Newton to Robert Hooke, 1676, as transcribed by Jean-Pierre Maury, 1992 22. KEPLERS PLANETARY LAWS, written by A. E. Davis, http://www-history.mcs.st-and.ac.uk/HistTopics/Keplers_laws.html 23. Spinoza and Jefferson, The Teaching Community, http://teachingcompany.12.forumer.com/viewtopic.php?t=2147 24. A History of Mathematics, written by Carl B. Boyer, 1985 25. The History of Computing Project, Blaise Pascal, http://www.thocp.net/biographies/pascal_blaise.html

Walmart Profile

Organizational Profile: Wal-Mart Wal-Mart is the largest retailer in the world. The position Wal-Mart holds gives the company a large responsibility to contribute to the community that supports the stores. As an organization Wal-Mart owes its success to the stake holders of their business. Wal-Mart requires the community to continue business operations, Trevino and Nelson state that â€Å"a major stakeholder in business must be the communities of which corporations and other organizations are a part† (2011). Wal-Mart must consider the community happiness with their business decisions to remain profitable.Wal-Mart gives back to the community in several ways including donations to charitable foundations and by offering affordable grocery good to impoverished areas. According to Feedingamerica. org Wal-Mart gave â€Å"$2 million donation to help food banks save money on their energy bills† (2011). The donation allowed 16 food banks to improve their efficiency and provide m ore services to the community for the same amount of funding. Wal-Mart has also received the â€Å"MOWAA Corporate Friend of the Year† (Meals on Wheels Association, 2012) award for its donation to the organization. Wal-Mart has proven that it contributes to the communities it supports.The contributions, when compared to Wal-Marts yearly profit, of â€Å"$16,389 million in FY2011† (Datamonitor, 2011, p. 4) seem miniscule. The true test of adequacy of Wal-Marts contribution to the communities it operates in will be if it can remain in business for the long term. According to â€Å"Paymar Communications†, â€Å"Wal-Mart also has been going through a major transformation and committing itself to progressive policies in the realm of corporate social responsibility, particularly when it comes to the products it sells and major efforts to promote energy conservation† (2010, paragraph two).Wal-Mart, as the biggest retail store in the United Sates, has a social re sponsibility to provide quality products to the customers. In 2010, the company took the initiative to reduce unhealthy levels of salts, fats, and sugars in their products. They also decreased the prices of healthy foods to encourage customer to purchase them. Wal-Mart also developed a seal to place on healthier foods with nutrition information rating the levels of sodium, fat, and sugar content (â€Å"Paymar Communications†, 2010). The leading factor in Wal-Mart taking this initiative was to reach customers with an interest in a healthy lifestyle.With obesity on the rise across the nation people are becoming more conscious about what they eat and offering healthier foods, organic products, and enhanced labels on the foods will give the customer more options to choose from within the store. By reducing the prices of these healthy products Wal-Mart can increase their customer base, specifically those with an interest in a pursuing a healthy life style. Additionally this will e nsure Wal-Mart is leading their competition and could be the benchmark for such companies as Target, K-Mart, Sears, and Costco.The factors influencing the organizations strategies are contingent on the need, not only of the customers, but also Wal-Mart’s increase in sales and profitability. Wal-Mart wants to ensure they not only have healthy customers but also that the organization will garner major gains in sales from these changes. The â€Å"Paymar Communications† (2010) website also states that through forging these initiatives, Wal-Mart has figured out a formula to be a good corporate citizen, give consumers superior bargains and still make tons of money; $14. billion in profits in 2009. It is the consummate win-win-win for the organization and customers alike. Wal-Mart’s social initiative is to help stop hunger in the communities in which it operates. Also Wal-Mart is providing healthful products to gain health-conscious customers and to help their customer s maintain a healthy lifestyle at a low cost. References DATAMONITOR: Wal-Mart Stores, Inc. (2011). Wal-Mart Stores, Inc. SWOT Analysis, 1-13. Paymar Communications. (2010). Retrieved from http://paymarcommunications. om/the-united- states-of-wal-mart-and-corporate-social-responsibility/ Trevino, L. K. , & Nelson, K. A. (2011). MANAGING BUSINESS ETHICS Straight Talk About How To Do It Right (5th ed. ). Retrieved from The University of Phoenix eBook Collection database. Walmart Foundation Gives $2 Million to Help ‘Green' Food Banks. (2011). Retrieved from http://feedingamerica. org/press-room/press-releases/walmart-green-initiative-2011. aspx We Are Meals On Wheels Association of America. (2012). Retrieved from http://www. mowaa. org/Page. aspx? pid=600

Getting the Best Topics for Comparison and Contrast Essay

Getting the Best Topics for Comparison and Contrast Essay Always have another look at your essay when you have finished. You may also order a fully written compare and contrast essay and alleviate the quantity of work you must do. You must be very careful as you pick a compare and contrast essay. If you don't, read how to compose a compare and contrast essay. Nowadays it is quite difficult to locate a trustworthy essay writing service. Before writing a comparative essay, you first have to arm yourself with the information you want. You may use the suggested topics as inspiration for your own, or you may simply opt to write about one you enjoy the most. Picking out essay topics is most likely one of the most difficult things about writing an essay, whatever sort it is. In the event you don't understand how to begin work on your essay, your best option would be the conventional hook technique. The very first thing you ought to do is identify the kind of compare and contrast essay which you are handling. The sole thing you must remember about picking the topic is it has to interest you. The first thing which you have to do before you get started writing is to pick an acceptable topic to write about. When you have the title and aspects to compare but without a definite number of criteria to cover similarities and differences, you've got to brainstorm. When you begin brainstorming and researching the subject, the odds are high you'll locate a wide collection of differences and similarities. Get the Scoop on Topics for Comparison and Contrast Essay Before You're Too Late If you think differences as opposed to similarities are somewhat more important for your essay, you should wind up with stressing differences, and vice versa. There is a range of interesting compare and contrast essay topics you'll be able to pick from you can describe laws you want to create, to provide some alterations to already existing laws, or to suggest to cancel a law which you don't like for specific explanations. Read distinctive viewpoints, even if it's the case that you don't agree with them. Topics for Comparison and Contrast Essay Options If it's the first time you're likely to use our article writing service, you most likely have a great deal of questions. If you wish to be certain that you are in possession of an excepti onal topic for your essay, you may look through the overall directions of our topics and hunt to find out more online. It is possible to discover contrasting features in 1 country during various occasions or ages. If you want to obtain a complimentary example, there's no greater place to search than an expert online writing website. It's common to be assigned to compose essays in every area of study, not just when taking a composition class. It's possible to choose books from fiction and non-fiction categories but make sure whatever books you decide on, they are from the exact same category. There are lots of others persuasive compare and contrast essay topics regarding education if you are prepared to write about doing it. Students who excel in writing about such complex topic might have an opportunity to be enrolled into a number of the ideal Art universities to come up with their abilities and talent. Again, compare and contrast essays can cover just about any topic and can pop up in a number of subject areas. Frequent themes are alike. To make it simpler, our experts have divided a number of the best topics into 4 distinct categories. Picking out the appropriate topics might take a while in case you don't have a list of sample topics before you. Selecting the ideal topic necessitates time. You must research your topic to select three claims. Following that, you can't locate a best topic. A noticeable topic is critical. Lucky for you, there are many topics you could concentrate on when writing and it's all your responsibility to ascertain the precise topic that you would like to build on. The Importance of Topics for Comparison and Contrast Essay Introduction with the highly effective hook and thesis statement stays the same. It's possible to restate your thesis statement and point out a number of the arguments used over the whole essay that backs it up. You should research the selected topic and discover facts to contradict your primary thesis. In the introduction paragraph, mention your thesis statement plus a concise explanation of the most important topic which you will explore. You only need to order a very good essay from experts with the greatest academic degrees in a number of fields. Also, there are plenty and plenty of resources to use to compose you r essay, so there shouldn't be any issue with your research. It's possible to write literature review in which you reveal your sources and the way they helped in your discussion. When it can be simpler to write on a well-known topic, with a little bit of research and organization, a writer can make an essay on an assortment of subject areas if assigned.

Homelessness The Current Social Welfare Issue - 1339 Words

Social Welfare Issue: Homelessness Written by: Laura Westra The current social welfare issue that I chose to explore is homelessness within Canada. First, this essay will give a brief description of homelessness. Secondly, it will describe how homelessness relates to some concepts and theories that we learned in this course Introduction to Social Welfare, it will also look at what may have happened to people facing homelessness one hundred years ago, next we will explore a possible solution to this social issue and where this fits within the political ideologies. Finally, I will give a brief explanation on whether social media is an effective tool for bringing about awareness regarding social issues. I believe homelessness is one of the most pressing social issues that Canada is facing, there are a number of diverse reasons for people living like this and the homeless population is more or less invisible to the rest of society. 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