Example research essay topic: Straight Lines Great Impact - 2,004 words

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Euclid's Elements Introduction to the works of Euclid and Euclid's Life Euclid's name is often associated with geometry. Euclid's Elements is one of the most influential and important scientific works in the history of mathematics. This work had served as the fundamental basics for the vast majority of geometrical teachings for the past 20 centuries. Euclid has made great impact on mathematics and set the standard for logical structure for works in the field of geometry.

Unfortunately, there is almost no information about Euclid's life. There are few facts indicating that Euclid lived during the reign of great Ptolemy I (306 - 283 BC), was well known in Alexandria, and established a mathematical school (Mueller). The historians mention that Euclid was a forthright, fair and kind person, who was eager to learn new things and spent all his life learning and teaching mathematics. Although he is most known for his Elements, there are other works written by Euclid, such as Optics, On Divisions of Figures, the Data, Phenomena, and some others. Similar to Euclid's Elements, these works have basically the same logical structure, definitions and thoroughly proved propositions. Yet, Euclid's Elements is the most important and the most influential work.

Euclid's Elements comprises of thirteen books, where the mathematician presents the elementary Greek knowledge about geometry of his days. This collection of books includes constructions and theorems of plane and solid geometry, the theory of proportions, number theory, the theory of incommensurable's and commensurable's, a type of geometrical algebra and other important information. Basically, Euclid's Elements is considered a set of theorems understanding of which leads to knowledge of the rest in the world. Euclid's Elements is a set of books in which the scientist has collected all theorems, concepts and fundamental knowledge constituting the foundation of ancient Greek mathematics.

Although this was not the first attempt to create such a textbook (as, for example, Hippocrates of Chios and other scientists also tried to write such fundamental work), Euclid's Elements was unanimously acknowledged to be the best and the most important collection of theorems (Investigating Euclid's Elements (essay review) ). Before examining the contents of this work and providing deeper insight into one of the most books in the Elements, it is important to mention what is not contained in Euclid's Elements. Although the mathematician widely uses the concept of rectilinear area, there is no specific formula explaining how to calculate the area of any figure (Gould). To a certain extent this can be explained by the fact that the ancient Greeks made differences between the art of calculation (or logistic) and arithmetic (known as number theory). Therefore, Euclid's Elements contains no logistic. Also, as far as Euclid bases his reasoning, calculation and geometry on straight lines, points and circles, his Elements contain no explanation or mentioning three most famous problems of Greek mathematics doubling the cube, squaring the circle and trisecting the angle.

Thirdly, it should be also mentioned that conic sections also are not to be found in Euclid's Elements, because they were still thought to be in the realm of higher mathematics during those days and Euclid treated conics in his separate work. Euclid's Elements Book I Although Euclid's Elements comprises of thirteen books, Book I is, by far, the most important book due to a number of reasons. In this book, the mathematician provides definitions to the fundamental terms of plane geometry, including line, point, angle, surface, figure and others. Although most of Euclid's definitions are undoubtedly logical and valid, there are few definitions that became subjects of heated debates either for their historical importance of for their originality.

For example, Euclid defines line, point and surface in a way that is significantly different from the definitions provided in the earlier geometry textbooks. For example, according to Aristotle, the definitions that were provided well before Euclid were defined in interrelation to each other; to put it differently, the prior definitions were defined in terms of the posterior ones. In other words, a point was defined as an extremity of a line; in its turn, a line was defined as an extremity of a surface, while the surface was defined as an extremity of a solid. Aristotle considered the way definitions were provided made them to be unscientific. In contrast to these unscientific definitions, Euclid tried to defined each term as an independent term that is not defined in terms of the posterior one (An invitation to read Book X of Euclid's Elements). For example, according to Euclid, the point is defined as something that which has no part' (I.

Def. 1), a line is defined as a breadth less length' (I. Def. 2), while a surface is defined as that which has length and breadth only' (I. Def. 5). After Euclid provided definitions to each term, he then turns to other definitions to find interrelation between them. There is another important definition that has significance for the history of mathematics.

In Book I, Euclid provides definition of parallel straight lines: Parallel straight lines are straight lines which, being produced infinitely in both directions, do not meet one another in either direction (I. Def. 23) This definition is basically, the same as the definition provided by Aristotle and is very important for the parallel postulate (I. Post. 5) that will be mentioned later in this research paper. Finally, it should be also mentioned that although Euclid provides definitions of such terms as rhomboid (that having opposite sides and angles equal to one another which is neither right-angled nor equilateral (I.

Def. 22) ), rhombus (that which is equilateral but not right-angled (I. Def. 22) ), and oblong, he never uses them in the Elements. According to some researchers, these definitions were taken from his earlier scientific works on geometry. In his Book I, Euclid provides very important postulates, axioms and theorems that are worth mentioning in this research paper. Each postulate in Euclid's elements is an axiom a statement that should be accepted without proof. Most of the postulates are constructions.

There are five postulates of Euclidean geometry: Let it have been postulated to draw a straight-line from any point to any point. And to produce a finite straight-line continuously in a straight-line. And to draw a circle with any center and radius. And that all right-angles are equal to one another. And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side). (Euclid, p. 7) In his first three postulates, the mathematician makes assumption concerning the existence of points, lines and circles.

This assumption is important, because, as it was already mentioned, by defining these geometrical objects, Euclid did not imply their existence, and the existence of other objects were proved later in his propositions. The first postulate can also be treated as the attempt to assert the uniqueness of straight line between two given points (Taisbak). Similar to it, the third postulate can be examined as assertion of the infinite extent of space and the continuity. The radius of the circle can be indefinitely large or small, implying that there is no maximum or minimum space between two points; therefore, the space is continuous. In the past, the 4 th and the 5 th postulates were considered theorems that could be mathematically proved.

According to the fourth postulate (And that all right-angles are equal to one another (Euclid, p. 7) ), the right angle is a determinant magnitude and all other angles can be measured against it. The fifth postulate is obviously one of the most interesting postulates, as it has been called the one sentence in the history of science that has given rise to more publication than any other. (Gould, p. 284) This postulate was considered a theorem due to its complexity and length, and mostly because its converse was a theorem that was proved by the mathematician (I. 17). After postulates Euclid presents common notions, or axioms (for example, things equal to the same thing are also equal to one another (Euclid, p. 7) ). First 26 propositions are all based on Euclid's four postulates and are mostly related to congruence of triangles and other geometric figures. Propositions on content triangles are also known to todays geometry as Side-Angle-Side (Proposition I. 4 if two triangles have two sides equal to two sides, respectively, and have the angle (s) enclosed by the equal straight-lines equal, then they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles suspended by the equal sides will be equal to the corresponding remaining angles (Euclid, p. 8) ), Side-Side-Side' (Proposition I. 8 if two triangles have two sides equal to two sides, respectively, and also have the base equal to the base, then they will also have equal the angles encompassed by the equal straight-lines (Euclid, p. 14) ), Angle-Side-Angle' and Angle-Angle-Side'' (both in Proposition I. 26 if two triangles have two angles equal to two angles, respectively, and one side equal to one side in fact, either that by the equal angles, or that suspending one of the equal angles then (the triangles) will also have the remaining sides equal to the [corresponding] remaining sides, and the remaining angle (equal) to the remaining angle (Euclid, p. 28) ). In Proposition I. 7 Euclid proves the uniqueness of a triangle with given sides, and in Proposition I. 22 the mathematician shows how to construct a triangle having any three given lines.

Proposition I. 32 proves the fact that in any triangle, (if) one of the sides (is) produced (then) the external angle is equal to the (sum of the) two internal and opposite (angles), and the (sum of the) three internal angles of the triangle is equal to two right-angles (Euclid, p. 34) ). The Proposition, also known as the Pythagorean theorem, is the most famous Euclid's proposition. According to it, in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). It can also be written as an equation: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c) In conclusion it can be said that Euclid's Elements are still considered a cornerstone of geometry, and a real masterpiece in the application of logic to arithmetic's and mathematics. This great work made great impact on science and such scientists as Johannes Kepler, Nicolaus Copernicus, Bertrand Russell, Galileo Galilei, Sir Isaac Newton, Baruch Spinoza and many others were influenced by Euclid's Elements and used this knowledge in their own works. Abraham Lincoln always kept a copy of Euclid's Elements and studied it very often.

The great success of Euclid's Elements is mostly explained by its relatively simple and logical presentation of the mathematical knowledge that was available to Euclid at his times. Although this work is very ancient, it still has great impact on modern geometry books and remains the cornerstone of modern mathematics. Works Cited "An invitation to read Book X of Euclid's Elements. " Historia Mathematica 19 (1992): 233 - 264. Gould, S. "The origins of Euclid's Axioms. " The Mathematical Gazette 1962: 284. "Investigating Euclid's Elements (essay review). " British Journal for the Philosophy of Science 34 (1983): 57 - 70. Mueller, Ian. Philosophy of mathematics and deductive structure in Euclid's Elements.

Cambridge, Mass. : MIT Press, 1981. Taisbak, C. "Did Euclid's circles have two kinds of radius. " Historia Mathematica 26 (1999): 361 - 364. The Elements. 15 July 2009 < web >.

Research essay sample on Straight Lines Great Impact