Euclid and the Birth of Euclidean Geometry
The ancient Greeks have contributed much to the development of the Western World as we know it today. The Greeks questioned all and yearned for the answers to many of life’s questions. Their society revolved around learning, which allowed them to devote the majority of their time to enlightenment. In answering their questions, they developed systematic activities such as philosophy, psychology, astronomy, mathematics, and a great deal more. Socrates (469-399 BC) was an ancient Greek philosopher whose ideas mark the turning point in the history of knowledge and formal thought. Plato (428-347:348 BC) one of Socrates students founded the Academy. The Academy was key in spreading thought and knowledge because of it’s devotion to teaching the sciences. Aristotle (384-322 BC), Plato’s brightest student, founded Biology and is given credit for his accomplishments in varying fields. Out of all of the great Greek accomplishments which influence the world today, I chose the one which I believe is the most important, Euclidean Geometry and its effects.
Euclid (365-300 BC) is often considered synonymous with geometry. Euclid’s works have been so influential that they serve as the basis for most geometrical teachings for the past 2000 years. His works supercede all other works of its kind. Euclid’s interests in spatial knowledge lead him to detailed definitions, postulates, and axioms that are used today. Data is a collection of given measurements and postulates that Euclid collected. Data expresses that lines, angles, and ratios can be given in magnitude; rectilinear figures may be given in species or form; and points and lines may be given in position. Euclid’s 94 propositions state that when certain aspects of a figure are given, other aspects can be found by using concrete formulas. For example, proposition 66 states, “If a triangle have one angle given, the area of the rectangle contained by the sides including the angle has to the area of that triangle a given ratio.” Divisions of Figures consists of 36 propositions concerning the divisions of various figures into two or more equal parts in given ratios. Optics is an elaboration on Platonic thought stating that discrete rays cause vision, and that vision can be explained by geometry. Euclid states that, “Thing...
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...n Theorem. Eudorus of Cnides solved Pathagorases dilemma of incommensurable magnitudes with the theory of proportion. Plato the teacher of many, considered geometry as the model of certain reasoning. Euclid during the 3rd century compiled and edited existing ideas. Pappus used Euclid’s writings as the basis for trigonometry, which is recorded in Almagest. Altogether the Greeks’ formalized geometry started the basis for modern trigonometry and set the grounds for the algebra of today, without all of the great mathematical contributions the world would be much different.
The mathematics ideas of ancient Greece are used in every aspect of life. The ideas of Greek mathematicians can be seen wherever you travel. From simple things such as buildings, to complex computers and engineering of all kinds, it is evident that their influence is ever present.
I am very impressed with the extent to which the ancient Greeks have influenced not only history but also our future. Euclidean Geometry and mathematics derived from it are used daily all over the world bringing order to the construction and understanding of almost everything.
The Ancient Greek contribution ranged by the 1900-133 BC, however its influence on the Western Literate Society lasts to this day. As the Greeks expanded their empire, they spread their ideas to other countries, while also borrowing from other cultures. During this period of time, the Greeks made many significant and long-lasting contribution to our modern culture in Philosophy, Art, Democracy, Drama, Math, and Science. These givings of important ideas, inventions, and structures have had an extraordinary influence on the surrounding environment, society, and in the future. The essential contribution of Greeks to the Western Civilization are Democracy, Art, and Philosophy.
The ancient Egyptians and Babylonians discovered abstract Geometry. They developed these ideas that were used to build pyramids and help with reestablishing land boundaries. While, the Babylonians used abstract geometry for measuring, construction buildings, and surveying. Abstract geometry uses postulates, rules, definitions and propositions before and up to the time of the Euclid.
Hydraulic fracturing, also known as Fracking, is a strategic system of obtaining natural gas that could be as deep as several thousand feet underground. Basically a mixture of sand, water, and chemicals are pumped underground at extreme pressures until they hit shale which contains the natural gas and in return the gas is forced up out of the ground. Fracking has become a highly successful process for increasing natural gas reserves in the United States. This process also lowers the United States dependence on foreign gas and oil and it accounts for an estimated forty two trillion cubic meters of natural gas, which is said to be enough to keep the United States in motion for sixty five years. As good as the fracking system sounds, there is always the down sides like the fact that the chemicals used to obtain the gas are extremely toxic. These toxic chemicals are pumped into the ground and can spread into nearby water. If this happens, the water will itself become toxic and undrinkable by neither animals nor humans. There are a lot of pros and cons of fracking, but what many people over the last few years have tried to say is that human life and safety is more important than an easier and cheaper way of doing something.
Fracking is also not legal in some areas of the world. In fact, in 2001 France banned hydraulic fracturing due to environmental concerns (The Facts on Fracking). On top of that, in some towns, cities, and counties in the states of New York, West Virginia, Texas, Colorado, Ohio, and California they have made moratoria or even banns of hydraulic fracturing (The Fracking Facts).One might say that hydraulic fracturing is good because in the United States it is helping us become a more independent country. On the contrary hydraulic fracturing is hurting the country because fracking can cause earthquakes or tremors. In fact, in Oklahoma the biggest earthquake was caused by a hydraulic
Fracking is quickly becoming a debatable topic in our society today. The practice involves injecting fluid into the ground to fracture rock in order to release natural gas. It sounds like it would be a safe way to harness fuels in the earth’s surface, but it actually is a danger to our environment. Because of the dangers of fracking, what little fresh water remains on earth is being contaminated. It is also releasing toxins into the airs creating contaminated air and acid rain. Because of the many health and environmental dangers of fracking, it should be stopped immediately to help prevent more worldwide health issues down the road.
“The Greeks molded the mind of Western Civilization, if not the body and the culture” (PowerPoint #6). The Greeks had a very powerful impact on the entire world, making them a huge influence to Western culture. The Greeks showed their power over the world by their great intellect. Through their accomplishments in art, architecture, government, philosophy, education, and science they helped shape Western Civilization. They built strong and intricate buildings, allowing others to see throughout history the plan of the architecture used. The Greek government was one of the first to have a Democracy. Their ac...
It is no mystery that without the Ancient Greeks, math as we know it today would not be the same. It is mind blowing to think that people who had no access to our current technology and resources are the ones who came up with the basic principles of the mathematics that we learn and use today without any preceding information on the topic. One of the best examples of such a person is Archimedes. Not only did he excel as a physicist, inventor, engineer, and astronomer, but he is still known today as one of the greatest mathematicians of all time. His contributions to the field laid out many of the basics for what we learn today and his brilliance shocked many. Long after his time, mathematicians were still stumped as to how he reached the genius conclusions that he did. Nicknamed “The Wise One,” Archimedes is a person who can never be forgotten.
"The Foundations of Geometry: From Thales to Euclid." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 1. Detroit: Gale, 2001. Gale Power Search. Web. 20 Dec. 2013.
Euclid, who lived from about 330 B.C.E. to 260 B.C.E., is often referred to as the Father of Geometry. Very little is known about his life or exact place of birth, other than the fact that he taught mathematics at the Alexandria library in Alexandria, Egypt during the reign of Ptolemy I. He also wrote many books based on mathematical knowledge, such as Elements, which is regarded as one of the greatest mathematical/geometrical encyclopedias of all time, only being outsold by the Bible.
Every idea has a start and a history that can be traced back in time. An incredible amount of these ideas and thoughts were started by great ancient civilizations. These ancient civilizations are the base of all modern knowledge. No ancient civilization has contributed more to this base than the civilization of Ancient Greece. The unique ways of ancient Greek agriculture have left a profound influence on the agriculture of today.
Even though Aristotle’s contributions to mathematics are significantly important and lay a strong foundation in the study and view of the science, it is imperative to mention that Aristotle, in actuality, “never devoted a treatise to philosophy of mathematics” [5]. As aforementioned, even his books never truly leaned toward a specific philosophy on mathematics, but rather a form or manner in which to attempt to understand mathematics through certain truths.
The Greek culture has had a huge impact on the history of the world. There is something Greek in almost everything, especially in the world’s architecture. Greece no longer had one king, so they focused on building temples for their gods. Architecture began small and plain but evolved into impressive pieces of art. As time passed from the Archaic period to the Hellenistic period, the people of Greece developed a type of formula for their buildings and their pieces of art.
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...