- Length: 4 pages
- Subject: Education - Mathematics
- Type: Term Paper
- Paper: #74585932

*Note*: Sample below may appear distorted but all corresponding word document files contain proper formatting

Euclid -- 323-285 B.C was a historical figure who taught at Alexandria in Egypt. There are three hypotheses revolving around Euclid's life. The first is that he wrote his magnum opus the Elements as also contributed a lot of other works. Another interesting hypothesis is that Euclid was a member of a group of mathematicians working at Alexandria with each one contributing to writing the 'complete works of Euclid', with the group engaged in writing books under the name of Euclid also after his demise. The third hypothesis is that Euclid of Megara lived roughly 100 years prior to Euclid of Alexandria. A team of mathematicians wrote the complete works of Euclid and took Euclid's name from Euclid of Megara. The proof surrounding the first hypothesis is significant that he wrote his magnum opus the Elements as also contributed a lot of other works. Scanty evidence is there that deny this hypothesis as it was readily accepted without any doubts for more than 2000 years. Apart from that, Euclid indisputably based the Elements of earlier works; hence it is quite possible regarding the reason behind the books written in different styles. (Euclid 323-285 B.C. Biography)

Although the first hypothesis is satisfactory, Euclid needed to have competent students who assisted him in his writing. Hence this would suggest hypothesis three, however, hypothesis three claims that different authors penned different books, but again there is little proof to suggest this. Based on the proof at hand, it will be assumed that hypothesis one is true. This leads to the fact that Euclid must have definitely studied in the academy of Plato in Athens to understand the geometry of Eudoxus and Theaetetus which is used a lot of times in his works. Since not a single work of Euclid's work contains a preface, it is not possible to gather much insights regarding any of his character from his works, therefore much knowledge about the life and the character of Euclid is absent. In this backdrop, Papus wrote that Euclid was " ... The more fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself." (Euclid 323-285 B.C. Biography)

Euclid contributed to mathematics through many books, but his most renowned one is his discourse on mathematics the Elements. The book Elements is a compilation of geometrical knowledge which is gained through meticulous proof that has come to be the core of teaching geometry for the last two hundred years. Euclid gathered his results from other sources; however he organized them into 13 Books with definition, proofs, and several lemmas. The treatise Elements is divided into 13 Books. Each one of them deals with a particular aspect of Geometry. Euclid relied on a major part of what be proves on the studies of other mathematicians, however he established a use of rigorous proofs which was novel to the study of mathematics and used by everyone who followed him. After the Holy Bible, the Elements are the most translated and studied among all books. (Euclid 323-285 B.C: Discoveries)

Euclid was known as the father of geometry. His book, the Elements contains the properties of geometrical objects and integers are attained from a small set of axioms, thus starting the axiomatic method of modern mathematics. Besides, Euclid's contribution has been on 'perspective', 'conic sections', 'spherical geometry' and 'quadric surfaces'. Even though several results in Elements started with earlier mathematicians, one of Euclid's important accomplishments was to put them in a distinct, logically coherent framework. Apart from providing some missing proofs, Euclid's text too includes sections on the number theory and 'three-dimensional geometry'. The geometrical system as explained in the book Elements was for a substantial period known as 'the' geometry. (Euclid: Wikipedia)

However, in the present era, it is pointed to as the Euclidean geometry to separate it from other non-Euclidean geometries that were discovered during the 19th century. These new geometries developed out of more than two millennia of investigations into Euclid's fifth postulate, which is one of the most-studied axioms in entire study of mathematics. Majority of these investigations tries to prove the comparatively intricate and presumably non-intuitive fifth postulate applying the other four, which is an accomplishment, if successful, would have demonstrated the postulate to be in fact a theorem. Whereas the Elements was used in the greater part of 20th century as a textbook on geometry and has been regarded an excellent instance of the formally precise axiomatic method, Euclid's treatment is unable to be comparable to modern standards of firmness, some logically useful axioms are not present and the explanations of primitive terms refer to spatial intuition. (Euclid: Wikipedia)

Euclid based his contribution in Book-I on 23 definitions like 'point', 'line' & 'surface', 5 'postulates' and 5 'common notions' both of which are presently known as 'axioms'. Some of the important postulates in Book I are (i) A straight-line segment can be drawn by connecting any 2 points. (ii) A straight-line segment can be expanded without limitations in straight line. (iii) A circle can be drawn by making use of the segment as radius & one end-point as centre when given a straight-line segment. (iv) Every right angle is congruent. (v) If 2 lines are drawn that have intersection with a third line in such a manner that the total of the inner angles on one side is not correct 2 right angles, then the 2 lines definitely should intersect one another on that side if expanded greatly. These basic principles show constructive geometry which Euclid together with his contemporary Greeks was eager upon. (Euclid's Elements: Wikipedia)

The first 3 postulates fundamentally mention the constructions one is able to perform with a compass and a straightedge of ruler which is not marked. The success of Elements is fundamentally because of its logical presentation of majority of the mathematical knowledge that is available to Euclid. Till the later part of 19th century, Elements was regarded as one of the best instances of a whole deductive structure: every of its component was believed to follow logically from its earlier components. Elements are even used in modern times as a sufficient example of the application of logic, and traditionally it has been heavily influential in a lot of spheres of science. European scientists Copernicus, Kepler, Galileo and particularly Newton were all inspired by Elements and applied their knowledge of it to their realm of work. Among the 5 postulates, the ultimate one so-called as 'parallel postulate' appears to be less obvious compared to others. (Euclid's Elements)

In mathematics, Euclidean geometry is the type of geometry on the plane or in types of three dimensions which is familiar. Mathematicians at times make use of the term to denote higher levels of dimensional geometries having similar properties. Euclidean goemetry at times refer to geometry in the plane that is also known as 'plane geometry' usually taught in high schools and Euclidean geometry in three dimensions is conventionally known as solid geometry. The traditional presentation of Euclidean geometry is as an axiomatic system, containing the proof all the true statements as theorems in geometry from a set of finite number of axioms. Presently, Euclidean geometry is normally constructed instead of axiomatized through analytic geometry. In case someone introduces geometry in this manner, one can then prove the Euclidean axioms as theorems in this particular model. However, this fails to possess the beauty of axiomatic method, but it is very concise. (Euclidean Geometry)

The Fifth postulate is one of the most studied axioms in the complete study of mathematics. Majority of the investigations focused around an attempt to prove Euclid's Fifth postulate from his other postulates which is now called as axioms, fundamentally demonstrating that the Fifth postulate was…[continue]

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