Introduction:
Thousand of years ago simple mathematics started with simple numbers being added and subtracted, then multiplied and divided. Now today, math has grown into complex functions and theories that are seen everywhere in different types of dimensions. Knowledge in math has greatly advanced since the beginning and Euclid was one of the many mathematicians that revolutionized math. His work with geometry was one of the most influential and he helped the math community take a leap forward.
Geometry in general has always interested me, especially since I love to sketch and paint. As an artist, during the sketching process of a painting it is necessary that shapes and basic principles of geometry are used to make a painting look real. Geometry has been always present in my life through art, so on my search for an Extended Essay I was looking at a topic related to math. I came across Euclid’s first five postulates and they interested me quickly as I read about them and their importance. Euclid was a very interesting person for he was more logical as compared to previous mathematicians, which I felt connected to, and I was compelled to start researching his postulates.
Euclid’s work in geometry influenced new theories in mathematics, especially the creation of non-Euclidean geometry. His work influenced people to continue working on growing the knowledge in the math community. Euclid revolutionized math to become more logic based as seen in his work in The Elements. His work is also used greatly everyday because geometry is necessary in many jobs and in making sure new and old mathematical concepts keep working. Since Euclid and his first five postulates helped advance mathematics and influence new fields of mathematics to be discovered, his work had a positive influence in the the math community.
Background
Euclid’s primary work was working with geometry, especially creating Euclidean geometry. Euclid wrote his work, findings and previous knowledge about geometry in the Elements, which consists of every definition, claim, proposition and postulate that help Euclidean geometry function. He used knowledge discovered about geometry by Thales and Pythagoras when writing his book. Euclid approached pure mathematics and geometry with the statement that all math should be proven with reasoning, without empirical measurements. (Neumann) Euclid started to get rid of empiricism in mathematics and make math be more logical and have reasoning (Raymound 174). Empiricism is the theory that all knowledge is derived from sense perception. Sense perception in math today is not used because Euclid was able to introduce new ways of using math, which involve logic and reasoning.(Neumann) The five postulates, introduced in The Elements, show Euclid’s theory of how math should be studied.
In The Elements he states the five postulates which are are important in having Euclidean geometry functions. These five postulates build from the first one and stop on the last one. The fifth postulate became known as the one with the most controversy because it doesn’t fit in with the other postulates. Since mathematicians noticed how different the fifth postulate was from the other they invested a massive amount of work to see why Euclid did it the way he did it. The work done on the fifth postulate also allowed for people to start working on new theories and mathematical fields, for example Non-Euclidean geometry.
Non-Euclidean geometry would be one of main mathematical fields that would arise from Euclid’s fifth postulate. The fifth postulate, also known as the parallel postulate, was trying to be proven by mathematicians, but Gauss started to look at the fifth postulate as false. In turn, he started a new field of mathematics, non-Euclidean geometry. It would be improved by mathematicians such as, Bolyai, Lobachevsky, Riemann and finished by Klein.(Connor 1996)
First Postulate:
Euclid based Euclidean geometry from five postulates that will always be true. The first of the five postulates states “Let it have been postulated to draw a straight-line from any point to any point”(Euclid). This postulate is explaining that the when you draw a two different points and connect them, there will be a straight line. Here is a picture to explain this postulate:
Euclid wrote this postulate to lay out the most important part of geometry, the line. The line us used everywhere in geometry, so Euclid saw that there must be a postulate for it. This postulate is seen in every one of the other postulates and is necessary for geometry to work. The line segment, described by the postulate, is marked as AB with two points A and B.
Second Postulate:
Next, the second postulate, Euclid stated “And to produce a finite straight-line continuously in a straight-line”(Euclid). This postulate, continuing off of the first postulate, states that any straight line created goes on infinitely and from that line there could be a finite line created with two points.
Third Postulate:
Now using the the the previous postulates, the third postulates states “And to draw a circle with any center and radius”(Euclid). Euclid states that with a line segment a circle can be drawn. The circle is created by a segment marking the center point and radius. This picture illustrates how a single segment creates a circle.
Fourth Postulate:
Onto the fourth postulate, Euclid states “And that all rights-angles are equal to one another”(Euclid). This postulate is meant to confirm that all right-angles are the same, so that other aspects of Euclidean Geometry are able to function correctly. This postulate will be used greatly in drawing shapes, so example squares and rectangles will always have four right angles no matter the direction it faces.
Fifth Postulate:
Finally, the fifth postulate states “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). This postulate is very different from the first four postulate, for it is not simple like the others and more difficult to comprehend. The image below provides an example of how the fifth postulate is defined.
The fifth postulate later became known as the parallel postulate for when the previous postulate is used there are two right angles created in the interior angles allowing the two lines to be parallel. This is not always true because there could be parallel lines created from angles that add up to 180 degrees, therefore this postulate seems becomes confusing in the way Euclid wrote his postulate. The image below shows how the lines become parallel.
The fifth postulate quickly would not be understood because it doesn’t directly state how two lines are parallel. Mathematicians became concerned because there was no direct postulate stating how two lines are parallel because in geometry parallel lines are very important. The fifth postulate was able show parallelism, but it is the most difficult of the five postulates. The fifth postulate, for Euclid, did not seem important because he did not use it for the first 28 propositions in The Elements. Other mathematicians would start trying to prove it from the fourth postulate because the other postulates worked like that.
The first four postulates work together and are able to create math that is seen in geometry, but the fifth postulate is not able to be easily carried out with the other postulates. The problem arises between the four and fifth postulate because there are no right angles mentioned in the fifth postulate. Mathematicians put extensive work in trying to prove the fifth postulate and starting with Gauss, a new field of mathematics started. Klein finished the work by Gauss and other mathematicians and he created Non-Euclidean geometry. This math field involved spherical geometry where all the math is done on a curve.
Own work done to prove the fifth postulate:
I have done my own work with logic and reason to try to prove the fifth postulate with the other four postulates. While working I managed to prove how you can use the postulates to show parallel lines. Below is the process which shows how I did it. First, I started with a single line segment, using the idea from the first postulate.
Secondly, I took the line segment and used it as a radius to create a circle using the ideas from the third postulate.
Then I make a cross in the circle with two lines and made sure that there are four 90 degree angles. This step uses the second and fourth postulates.
After, I drew a line through the points C and B in the circle.
Finally, I drew a line through points D and E. From this picture the lines CB and DE are parallel. This could be proven with angles and the two triangles should be equal.
The fifth postulate is seen, in my images, when the radii of the circle are used to create triangles. From the use of all the postulates I am able to prove that two lines are parallel with the creation of a circle between the lines. You could use this idea, I created to prove whether lines are parallel. So, you would need to find two lines that look parallel. In between the lines you will create an X symbol, similar to the one I used in my example. Then you have to make sure that angle c and angle e are equal, and that angle b and d are equal. If the triangles are congruent with the two strains lines being the hypotenuses, then the lines should be parallel.
This image uses each of the postulates to prove parallel lines. I started with two points and connected them with a line segment. With that line I created a circle. Inside the circle I created a right isosceles triangle with the angles 45’ 45’ 90’. This triangle uses the fourth postulate because it uses a right angle. Now the fifth postulate is seen in the triangle and the hypotenuse line. The two lines in the triangle meet in the middle of the circle forming a right angle. Now, another triangle is made by reflecting the first triangle across the center point. This triangle appears in in the bottom left quadrant of the circle. The two hypotenuses of the triangles should be parallel. This is my idea of how all these postulates are proven together, but each postulate could be used in many different ways to prove many theories. Examples of the postulates working together is shown in Euclid’s propositions section of The Elements. In my work I used Euclid’s work to prove that two lines are parallel, while other people.
Connections to today:
Over the years, many people have started to expand from Euclidean Geometry and create a new geometry that looks at geometry from different dimensions. Non-Euclidean Geometry was one of the most popular geometries to be worked on after Euclid’s work. It was started by Cayley in the late 19th century and later continued by B. Klein who finalized Non-Euclidean Geometry. They focused greatly on the parallel concept from Euclid because they needed to prove parallelism in their geometry where it was focus of hyperbolic and spherical geometry. Non-Euclidean Geometry is focused working with geometry with curved shapes, while in Euclidean Geometry the work is all done on flat 2-dimensional surfaces and solid 3-dimensional objects. Cayley started by looking at dimensions higher than that looked by Euclid and previous mathematicians. (O’Connor)
Euclid’s work of Euclidean Geometry greatly benefitted the math community because they were able to grow mathematics and expand it. Without Euclid’s work, there might not have been mathematicians looking it different dimensions and creating rules that allow for people to work in those dimensions. Mathematicians might not have been influenced to use geometry in a spherical settings involving curves. Euclid wrote the fifth postulate differently than the others postulate and this brought people together to figure out the reason behind Euclid. While working they are able to discover new fields of math and come up with different conclusions from Euclid’s findings.
Euclid was able to keep the math community growing and expanding their knowledge with his influential work with Euclidean-geometry and the Elements. Mathematicians were faced with a problem when Euclid released his work on Euclidean-geometry and many mathematicians found a problem with the fifth postulate. The fifth postulate was different from the others and it did not fit in. This brought in many mathematicians to study Euclid’s work and prove the fifth postulate from the other postulates. Work done by mathematicians such as Gauss, Bolyai, Lobachevsky, Riemann and Klein brought a new type of math called non-Euclidean geometry.(Norbert 12) Euclid was also able to get rid of empiricism in mathematics and change math to be done with lots of logic and reasoning in understanding math and solving problems. Euclid and the first five postulates of Euclidean geometry allowed for new ideas to flourish and the changed the way math is being done today.
Today Euclidean geometry is seen in many places, for it involves shapes, lines and angles. Geometry is that mathematical field that is seen in every other field and it is seen and used everywhere in our world. Euclid work from about two thousand years is still being used today to help people complete their jobs and help new mathematical concepts emerge because old owns stay true. The five postulates are seen mostly in jobs that require knowledge of geometry. Jobs such as architects, mechanical engineers, drafters, urban and regional planners, and surveyors use geometry extensively. For example, in architecture most of the work requires geometry and some physics to make sure that a building can be able to function and stay up. Architects would have to create blueprints that would be scales of larger buildings and structures. In most of these blueprints Euclid’s five postulates are seen. The architect would use lines and circles to mark out walls and edges of a floor. Right angles are indicated to show corners in rooms. The fifth postulate is harder to notice, but with the fifth postulate parallel lines are created. These parallel lines are seen frequently in blueprints because the architect would usually want to make his work look symmetrical.
Conclusion:
During my investigation I struggled in finding a way to show that I have done math, but as a kept doing research I felt that I could show that I understand Euclid’s postulates by using them together. My example shows how those postulates work together and that example could be used in many things today, for example it with Euclid’s postulates a circular base of a structure could be made. Since circles are seen nearly everywhere, Euclid’s postulates can be used together in many circumstances today. Each of the postulates can also be used in their own way in geometry. Euclid’s first five postulates influenced math mathematicians to go further and they also help geometry work today in many jobs, such as architects, mechanical engineers, drafters, surveyors, urban and regional planners, and many other smaller jobs. Euclid’s work is important in our lives and it has positively influenced the math community and the communities around it for new discoveries were found and they are used nearly everyday.
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