The Discovery Of Non Euclidean Geometry

your eyes to see this helped with the creation of geometry.A specific principle that Da

David Hilbert was a German mathematician whose research and study of geometry, physics, and algebra revolutionized mathematics and went on to introduce the mathematic and scientific community with a series of mathematical equations that have yet to be solved. Furthermore, his study of mathematics laid the groundwork for a variety of ongoing mathematic analyses, which continue to influence the world today.

Hipparchus was a greek astronomer, geographer, and mathematician born 190 B.C. in Nicaea and died in 120 B.C. Rhodes, Rhodes, Greece. Hipparchus is accredited as the inventor of trigonometry because of his discovery of the first table of chords and also because he's the only person with valid data of the discovery and usage of trigonometry. In order to calculate the rising and setting of zodiacal signs, Hipparchus brought to light the division of circles into 360 degrees and the calculation of chords by looking at the triangles (spherical triangles or triangles that made up a circle) differently. Hipparchus experimented putting all triangles to be within a circle and with the three points each touching the

Did you know that Pythagorean Theorem was actually created well before our generation’s time period? Yes, it seems that is pretty basic knowledge, but did you know it was actually created in B.C. and instead of A.D.? The brilliant mind to thank for such an achievement is the man who goes by the name Pythagoras. However the Pythagorean Theorem isn’t his only accomplishment, he was so advanced for his time, that he had many achievements. But to the best way to examine these accomplishments is to start at the beginning.

When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.

Geometry and mathematics were the cornerstone of Greek life. While the myths of the day were still acknowledged and, for the most part, accepted, it became less prevalent in society.

The renaissance architects and artists has already influence into the mathematical term of portion and perspective because their works have ticked the rivaled nature. As explain the article Galileo used mathematical in a equal skill to reveal that the underlying structure of physical space and emotion could be reduced to mathematical analysis. So in his analysis Galileo described that connecting physical space and real motion could be observed that a uniform change of his neo-platonic, mathematical world. Galileo was the most important person in Europe because he was the bridge for the the scientific revolution in that continent.

The author of Journey through Genius, William Dunham, begins this chapter by depicting how mathematics was spurred and developed in early civilizations. Dunham focuses primarily on the works’ and achievements’ of early Egypt, Mesopotamia, and Greece in this section. These ancient societies, as they developed, produced mathematicians such as; Thales, Pythagoras, and Hippocrates, who turned a basic human intuition for space and quantity into applicable everyday mathematics. The primary influences driving the development of early mathematics were the issues of growing civilizations, most notably counting commodities, taxation, and the division of land equally, rather than a pure desire for understanding that is seen in mathematics today. These influences culminated in the development of early arithmetic and geometry.

When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.

The Greeks and the Egyptians used triangles as early as 3500 BCE. They used these triangles as rules of thumb. They could apply these rules to specific applications. For example, the Egyptians knew that the 3:4:5 ratio was a right triangle. They could derive this because for them to create a right triangle the Egyptian land surveyors used a rope divided into twelve equal parts, creating a triangle with three pieces on one side, four pieces on the second side, and five pieces on the last side. The right angle was found where the three-unit side came together with the four-unit side. This was a very efficient way to create right triangles. It’s a mystery as to how the Egyptians came up with this, but this was later used by Pythagoras (c.571 -

2. Eratosthenes, a Greek scholar, wrote a book with a title meaning "Earth description", which is similar to our modern definition of geography. Although no copies of his book survived, we know from other authors that Eratosthenes supported the idea of a round earth. Eratosthenes also drew intricate maps of the known world and determined the circumference of the earth with surprising accuracy. Hipparchus, another Greek scholar, was the first to draw an imaginary grid over the Earth's surface to divide it and simplify finding specific locations. The Roman Empire contributed to ancient knowledge of geography as well; a collection of geographic study expansions by Romans was recorded by Ptolemy, who was yet another Greek scholar.

     Euclid's most famous work is his dissertation on mathematics The Elements. The book was a compilation of knowledge that became the center of mathematical teaching for 2000 years. Probably Euclid first proved no results in The Elements but the organization of the material and its exposition are certainly due to him. In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions, which are never used such as that of an oblong, a rhombus, and a rhomboid. This book first began the book by giving the definition of five postulates. The first three are based upon constructions. For example, the first one is that a straight line can be drawn between two points. These three postulates also describe lines, circles, and the existence of points and the possible existence of other geometric objects. The fourth and fifth postulates are written in a different nature. Postulate four states that all right angles are equal. The fifth one is very famous. It is also can be referred to as the parallel, the fifth parallel. It states that one and only one line can be drawn through a point parallel to a given line. His decision to create this

Euclid’s assumptions about his postulates have set the groundwork for geometry today. He provided society with definitions of a circle, a point, and line, etc and for 2000 was considered “the father of geometry.” His postulates proved to be a framework from which mathematics was able to grow and evolve, from two thousand years ago, till Newton and even to all our classrooms today.

According to Victor Katz in “A History of Mathematics (3rd Edition)” (Pearson, 2008), trigonometry developed primarily from the needs of Greek and Indian astronomers. But today, trigonometry and geometry were used on various fields such as: architecture, physical sciences, engineering, astronomers, medical imaging (CAT scans and ultrasounds), etc.

It has been said that Cartography originated thousands of years ago. Wall paintings believed to represent a maps of the heavens exist all around the world. The oldest maps that have been preserved on Babylonian clay tablets are from around 2300 B.C. Cartography as a whole was dominant in Greece as the subject was more advanced than other locations and so the spherical Earth was well known amongst Greek philosophers. (Aber, 2008). The reference-line principle was created by one of Aristotle’s disciples. Dichaearchus was this disciple’s name and he obtained this theory by placing an orientation line that ran from east to west. (Fuechsel, 2013)