Euclid: The Father of Geometry In our day and age, it is rare to read and research a person who has shaped such a fundamental

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1 Elaine Chen History of Mathematics Essay 1 Euclid: The Father of Geometry In our day and age, it is rare to read and research a person who has shaped such a fundamental subject in our lives without much information about who they are; in this case we have Euclid, the Father of Geometry. Although Euclid flourished around 300 B.C. he was believed to be a student of Plato and captured Greek mathematics in his books. Euclid is best known for his work Elements, which consists of 13 books and is one of the most influential textbooks of all time second only to the bible in the amount of editions published. The book replaced all previous text on geometry comprised of postulates, propositions, proofs, and definitions. Although Euclid did not invent geometry nor was the first person to study it, his mathematical proofs and presentation were strictly his work 2. Elements were based off of 10 assumptions. The first 5 were postulates stating that: I. To draw a straight line from any point to any other. II. To produce a finite straight line continuously in a straight line. III. To describe a circle with any center and distance. IV. That all right angles are equal to each other. V. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles 3.

2 The next 5 assumptions were axioms: 1. Things which are equal to the same thing are also equal to each other. 2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part 7. The purpose of these assumptions were to be used to help prove theories, propositions, proofs, and any other result he was seeking. These were not the only assumptions he used but the ones that he chose to emphasize on. These assumptions set the outline for all of the Elements books. Euclid organized the books by having the first six books pertaining to plane geometry, seven through nine dealt with number theory, book 10 dealt with Eudoxus theory of irrational numbers and the last three books dealt with solid geometry. Now the difference between other textbooks and Euclid s was that he made Elements clear and easy to understand. Theorems are stated, then they are proved 2. If you take a look at Proposition I-29 in Book 1 below, you can see a simple math figure that we have all seen at school in one point in time 1. We know that the straight line that intersects the two horizontal parallel lines creates equal angles between AGE & FHD, EGB & FHC, GHD & AGH. Proposition III-11 is another simple fundamental stating that if circles touch one another

3 internally, and their centers be taken, the straight line joining their centers, if it be also produced, will fall on the point of contact 1. Euclid made those and other propositions easy to understand especially during that time period in Greece. Although Euclid s Elements have been so influential for so long, there are people who have tried to prove Euclid wrong. Some people say that modern mathematics has surpassed Greek mathematics and therefore are inadequate for our time now. A. E. Meder argued that he did in fact attempt, unsuccessfully of course, to define everything 4, he didn t set forth properties that distinguished entities to be defined from all others in regards to A point is that which has no parts and A point is that which has no parts. Others have also argued that Euclid s worst offence was that he failed to mention any order of points on a line or even a concept of something in between. This means that if you have the figure below: the point B cannot be stated to be between A and C because it doesn t strictly follow the basis of Euclid s formulation of geometry 4. Even though Euclid had his critics, his work is still resonating all throughout the world.

4 Now why is all of this Euclid information important? All of the fundamental concepts that Euclid had in Elements contained work from other philosophers, mathematicians, and some of his own discoveries, all encapsulated into thirteen books, and somehow into our brains still today. In France today, students are to be able to transform figures in sixth grade, solve problems involving centers, orthocenter, projections, translations and others in eighth grade, while in ninth grade they are taught vector geometry all fundamentally based off of Euclid s geometry. In Japan, their most prominent form of teaching Euclid s geometry is to make geometry algebraic; this is also practiced in the United States 8. Although Elements was used as the main mathematics book in schools in Europe, West Asia, and America for two thousand years until the 20 th century 7 there are still many curious minds studying geometry from his books even if thirteen books seems excessive because there is no royal road to geometry 5. 1 Allen, Don. "Euclid." TAMU, 14 Feb Cobb, Alayna. "Euclid C B.C.E. WSU 3 Mastin, Luke. "HELLENISTIC MATHEMATICS - EUCLID." 4 Meder, A. (1958). What is wrong with Euclid? The Mathematics Teacher, 51(8), Retrieved from 5 Pas, Stéphanie Van Der. "THE NORMAL ROAD TO GEOMETRY: δή IN EUCLID'S ELEMENTS AND THE MATHEMATICAL COMPETENCE OF HIS AUDIENCE." The Classical Quarterly The Class. Q (2014): OneSearch. 6 Ji, Shanyu. Lecture 10. The Elements of Euclid University of Houston

5 7 Shuttleworth, Martyn. "Euclid, the Father of Geometry Greek Mathematics. " Explorable, 12 May Yunming, Zhou, and Zhao Zhe. "The Studying and Revelations of Euclid Geometry of Developed Countries." 2010 International Conference on Computer and Communication Technologies in Agriculture Engineering (2010). OneSearch.

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