Brittiny Tran MATH 4388- Dr. Ji Essay 1: The Evolution of Euclidean Geometry Geometry is ancient and universal. The history of geometry goes back to about nearly 4,000 years ago in Mesopotamia, Egypt, India, and China. The geometry that is well known today, started back in 600 BC, which was when Thales of Miletus brought logic and logical arguments into geometry. This later became known as Euclidean Geometry. Since Euclid was the only geometer to explain Greek geometry in a way that we can understand, he is often referred to as the “Father of Geometry1.” Euclidean Geometry is the geometry that ‘makes sense’ and the one that everyone is more familiar with. In fact, most of the geometry textbooks used in high schools were most likely based off his book, Elements. It is common knowledge that to construct any geometric shape, lines are necessary. In Euclidean geometry, lines are made up of actual Euclidean points whereas in other geometries they are made of vague ‘line stuff’. Euclidean Geometry is one of three geometries that are known as “The Big Three2.” The other two geometries are Spherical and Hyperbolic Geometries. These three have many similarities and differences. In fact, all three have a distance function and vertical angles are congruent. “The Big Three” are axiomatic structures. Therefore, they have all undefined terms, axioms, theorems and definitions. The Angle-Side Angle Theorem applies, and Side- Angle-Side is an axiom for the space of the geometries. ______________________________ 1. Martyn Shuttleworth, Explorable.com 2. Gerard A. Venema, Foundations of Geometry Second Edition However, there are numerous properties and characteristics that sets each of these geometries apart from each other. For example, the measure of an exterior angle is the sum of the measures of the remote interior angles is only true in Euclidean! The statement, “the Pythagorean Theorem is really a theorem about side lengths3” only applies to Euclidean geometry. The only geometry that is unbounded is Euclidean. Euclid proposed that “things equal to the same thing are equal4”. Euclid’s Elements, may be “the most influential non-religious book of all time.5” Shuttleworth states that Elements is organized by subjects that pertain to “every area of mathematics developed by the Greeks.” These ranged from basic geometry to whole numbers. Euclidean Geometry is “the most typical expression of general mathematic thinking.6” Euclid had begun with explaining some undefined terms such as a point and line. He defined a point as “that which has no part.4” He considered a line as “a length without breadth.4” From there, he started to define what angles and various shapes were such as circles and triangles. Euclid explained angles as the incline of two straight lines; circles were “plane figure(s) consisting of all points that have a fixed distance (radius) from a given centre.4” For more aspects that required more logical thinking, Euclid provided five principles that would be the basis otherwise known as axioms. The first axiom or postulate states that a line joins two points together. The second says that a straight line can be growing infinitely. The third axiom defines that a circle is made when two things are given: a center point and distance for the radius. The ______________________________ 3. Gerard A. Venema, Foundations of Geometry Second Edition 4. Benno Artmann, Encyclopaedia Britannica, inc., https://www.britannica.com/science/Euclidean-geometry 5. Martyn Shuttleworth, Explorable.com next declares that “all right angles are equal.6” Lastly, “a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.6” In other words, if two lines don’t intersect, then they are parallel lines. This fifth axiom or postulate later became known as the parallel postulate. Now, even though, Euclid is considered the “Father of Geometry,7” there were other geometers that opposed some of his teachings. The ancient geometers had a deeper, “more advanced understanding of geometry.8” They believed that geometry was “tied to the physical world.8” On the other hand, the ‘more modern’ geometers are complacent and more receptive to the newer methods/teachings. They typically think that Euclid ‘is right.’ Nowadays, geometry and mathematics is more abstract due to the axiomatic systems. These systems help us to comprehend the basis of each geometry such as “The Big Three8”. Throughout history, it has been discovered that both geometry and mathematics utilize logic and can be applicable to real-world situations. Out of “The Big Three8,” Euclidean geometry is hands down the easiest to grasp. In the class textbook, it is stated that “Euclidean geometry has been regarded as the unique way to observe our universe.9” However, since then, the “Geometry Explosion8” occurred in the 19th Century and “Non-Euclidean Geometry” was introduced. ______________________________ 6. Benno Artmann, Encyclopaedia Britannica, inc., https://www.britannica.com/science/Euclidean-geometry 7. Martyn Shuttleworth, Explorable.com 8. Gerard A. Venema, Foundations of Geometry Second Edition 9. Dr. Shanyi Ji, History of Mathematics, p. 68 Works Cited Artmann, Benno. “Euclidean geometry.” Encyclopaedia Britannica Online. Encyclopaedia Britannica inc., 2016. Web. 15 Sept. 2018. Shuttleworth, Martyn. (May 12, 2010). Euclid, the Father of Geometry. Retrieved Sep 15, 2018 from Explorable.com: https://explorable.com/euclid Venema, Gerard A. Foundations of Geometry Second Edition. 2012 .