Euclid’s Fifth Hypothesis Term Paper Samples
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The first four postulates of Euclid were considered as a given and is easily seen and proven. They were assertions that are readily understandable. Euclid’s fifth postulate however is considered as the most different and difficult since its proof is not as obvious. In fact it is so different that it is the reason that paved the way to non-Eucledian geometry. Many mathematicians such as Bolyai, Lobachevsky, and Gauss tried to prove it to be considered as a theorem and to erase the necessity to assume it, but certain assumptions were needed in order to fully understand and explain what it says (Joyce).
Euclid’s postulates are found in his book Elements, specifically beginning with Proposition 27. Although the book does not explicitly divide the chapters into how it is taught and understood today, there was a distinct difference from Proposition 27 from the previous ones. From this proposition, to the 34th discusses parallelism which is the reason why the fifth postulate is often called as the “parallel postulate” (people.whitman.edu).
The Fifth Postulate
“If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.”(Weisstein)
Figure 1 presented below graphically shows the workings of this postulate. Given two lines, (l1 and l2) and another line intersecting these two lines (l) whose inner angles on one side (α,β) has a sum that is less than two right angles (α+β<1800) then extending the line on the side of those angles will lead to the two lines intersecting.
Figure 1. Illustration of the Fifth Postulate
The Fifth Postulate and its Corresponding Propositions
As stated a while ago, the fifth postulate is where prepositions 27 to 34 are banked upon. This chapter discusses each of these propositions as it shows how Euclid viewed parallelism. Proposition 27 (illustrated in Figure 2) states that given two straight lines (AB, CD) and a straight line that meets both of them (EF) and its alternate angles (E and F) are equal, then the previous lines are parallel.
Figure 2. Illustration for Proposition 27
Proposition 28 (as illustrated in Figure 3) states that given a straight line (EF) that meets two straight lines (AB and CD) makes an exterior angle (G) equal to the opposite interior angle on the same side (H), or if it makes the interior angles on the same side equal to two right angles (180º), then the two straight lines are parallel. Proposition 29 does not need its own figure since it simply state the alternative of Proposition 28. If two straight lines (AB and CD) are parallel, then a straight line (EF) that meets them makes the alternate angles equal (GH), it makes the exterior angle equal to the opposite interior angle on the same side. Additionally, it also makes the interior angles on the same side equal to two right angles.
Figure 3. Illustration for Proposition 28
Proposition 30 (as illustrated in Figure 4) simply states that given two straight line, say AB and CD, that are parallel to another straight line, EF, leaves the first two lines parallel to each other.
Figure 4. Illustration for Proposition 30
Proposition 31 (as illustrated in Figure 5) allows to draw a straight line that passes through a given point parallel to a given line. In the figure below take Pt. A as the given point and BC as the given line. In order to create line EF, a straight line must be created from Pt. A to Pt. D which at the same time makes angle DAE equal to angle ADC which from proposition 27 assures that the lines are parallel.
Figure 5. Illustration for Proposition 31
Proposition 32 (as illustrated in Figure 6) states that “If one side of a triangle is extended, then the exterior angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to two right angles” (Math Page). From a triangle extend one side (CE) which will form a line that is parallel to AB. Since AB and CE are parallel it can be said that angles ABC and ECD are equal. At the same time angle ACD will equal the sum of angles ABC and CAB.
Figure 6. Illustration for Proposition 32
Propositions 33 and 34 (illustrated in Figure 7) may be explained in the diagram below. The first proposition simply states that if two parallel lines that are joined on its edges by two more straight lines then the lines on the sides are themselves equal. This serves as the essence of a parallelogram. The final proposition states that given a parallelogram the opposite sides and angles are equal and the diagonal line bisects the area, cutting the whole shape in half.
Figure 3. Illustration for Proposition 33 and 34
Euclid’s fifth hypothesis was an obstacle that even he had a hard time to tackle. The need for it however cannot be questioned for it completes the idea of his geometry. Euclid’s work served as an important framework to how geometry is currently viewed. The applications of these concepts go far and wide which is evidenced by the continuous need to teach these concepts for it still is very much relevant.
Weisstein, Eric W. "Euclid's Postulates." From MathWorld--A Wolfram. Web. 25 March 2015 <http://mathworld.wolfram.com/EuclidsPostulates.html>
“The Fifth Postulate”. Web. 25 March 2015. < http://people.whitman.edu/~gordon/wolfechap2.pdf>
Joyce, David. “Euclid’s Elements: Book One”. Clark University. 2003.Web. 25 March 2015. <http://aleph0.clarku.edu/~djoyce/elements/bookI/post5.html>
“The Math Page”. Web. 25 March 2015. < http://www.themathpage.com/aBookI/propI-33-34.htm>
Norton, John. “Euclidean's Fifth Postulate”. HPS University of Pittsburgh. 1 January 2013. Web. <http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_fifth_postulate/index.html>
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