Ptolemy's Theorem Essay

This shows Ptolemy based his work off of other scholars that mostly look at the church for things whereas Copernicus did his own research and used facts to back up his theory. The ideas of Copernicus have changed the way people thought about the nature of man and man's place in the universe because they no longer thought earth was the center of the universe. Another example of someone conducting their own experiment is Andreas Vesalius on the human body. Before his theory of the human body people believed that each sign of the zodiac controlled a certain part of the body. An example is the constellation called Aries the Ram controlled the head.

He used logic and reasoning to investigate the nature of the universe. Ptolemy’s model helped get people thinking about the universe being eternal, finite, and spherical. This document is most definitely not reliable because the conclusion was made without having all the needed information, and it was later disproved. A more accurate view could come from someone who does in depth studies of how the universe is actually laid out. As seen in Document 11, A Greek doctor by the name of Hippocrates became one of the first people that taught his belief that illnesses only had

Greek mathematical and scientific ideas have also been a large contribution. Euclid, a Greek mathematician, wrote many theorems. His theorems have been very substantial because they are included in today’s mathematical problems. One of his theorems include “If two straight lines cut one another, the vertical, or opposite, angles shall be equal.” This specific theorem is used in proofs and it proves that all vertical angles are equal. Hippocrates was a Greek physician who created the Hippocratic Oath. This oath

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

Given ABCD is a square and rhombus EFGH is inscribed in the square. We want to prove that EFGH will be a square.\\

His model said that the planets moved not in circles around the sun, but in ellipses and the mathematics was proved using three laws:

One of Thales’ most renounced findings include his discovery in geometric studies in the area reading the rules of triangles. He came to the conclusion that if the base angles of an isosceles triangle are equal, the sum of the angles of a triangle are equivalent to two right angles. With the application of “geometric principles to life situations, Thales was able to calculate the height of a pyramid by measuring its shadow, and the distance of a boat to the shore, by using the concept of similar triangles” (pg. 5, Muehlbauer). Realizations such as these helped shape the beginning for the formation of natural law based on observations of the world through explanation.

Review the Ptolemaic model and the Copernican model. Also, give a lecture on Nicolaus Copernicus and Ptolemy as individuals.

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 made several inventions, most notably in the subject of math, which are still studied today and taught in school. Mathematician Euclid is often credited as the “Father of Geometry” for all his work and studies in this subject, which are compiled in his books called The Elements. He organized known geometrical statements called theorems and logically proved all of them. He proved the theorem of Pythagoras (another Greek mathematician), which stated that the equation (c2 = a2 + b2) is true for every right triangle.

The discovery of non-Euclidean geometry is credited to nineteenth-century mathematicians Carl Friedrich Gauss, Nikolai Ivanovich Lobachevsky, and János Bolyai because they are first to recognize that the negation of Euclid’s Fifth Postulate as an axiom produced another geometry that was as rich and solid as that of Euclidean geometry (Venema, 2012). However, several concepts of Hyperbolic geometry were already known by that time, it just was not labeled as Hyperbolic geometry (or non-Euclidean geometry) because Hyperbolic geometry was not yet “discovered”. Euclid’s work was so distinguished that it was accepted as the standard so people did not think it was possible for other geometries to exist. All in all, the discovery and growth of Hyperbolic geometry first began with several mathematicians trying to prove that Euclid’s Fifth Postulate could be eliminated as an axiom.

41), I.26 (pg. 43), I.27 (pg. 44), I.29 (pg. 45), and I.32 (pg.46). These propositions set the stage for the ‘great theorem’ of chapter 2 “Euclid’s proof of the Pythagorean Theorem.” This proof is important because, until this point in history, a formal written explanation and proof was not existent despite its wide acceptance as truth. Proposition I.47 states, “In right-angled triangles, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.” Euclid tackled the great proof by using geometry to describe the equation, c^2= a^2+b^2, as seen in Figure 4. The ultimate goal is to show that the area covered by the two smaller squares sum up to the area of the third square. Initially, we pose that triangle ABC is right by design. Then, by proposition I.14, GAC must be a straight line. By examining the ΔABD and ΔFBC, we see that they have sides of (AB) ̅=(FD) ̅ and (BD) ̅=(BC) ̅ since they are sides of squares. Next, since angle ∠ABD=right angle+∠ABC and ∠FBC=right angle+∠FBA. Using postulate I.4 and common notion 2 we know ∠ ABD=∠ FBC. Consequently, Euclid could now show that ∆ABD=∆FBC by the side-angle-side triangle congruence. Now Euclid applied his postulate I.41 to relate the area of the squares, to the area of a triangle that shares the same base.

     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.

Trigonometry and geometry affects us even in the most essential point in our lives. It helps us understand different aspects in our lives and improving the quality of life.