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Pythagoras and his contributions
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Greek Geometry
Although the original roots of geometry can be traced to the Egyptians, the Greeks built on most Egyptian theories that we use today. Greek astronomy and Greek geometry were both used in order to answer many difficult questions of the time. Without geometry, the study of astronomy would have been almost impossible, and vice versa. Even though many Greek theorems and principles were later built on by geniuses such as Einstein and Lobachevsky, the basis still remains the same.
The development of Greek geometry is said to be started by Thales of Miletus. Thales came from Egypt with a number of geometric principles that the Greeks were able to use for practical purposes. He lived towards the beginning of the sixth century B.C, and has been credited with many geometric theorems. Some of the most important theorems developed by Thales included:
- If two triangles have two angles and one side is respectively equal, then both triangles are congruent to each other.
- Angles at the base of any isosceles triangle are equal.
- If two straight lines intersect, then the opposite angles formed are equal.
Thales also did much work with the height of pyramids by measuring the height of the pyramid's shadow only at a specific time of the day. While most of his theorems were proven, some that were not pertained to a ship's distance from shore and the bisector of a circle. His discoveries led to the formation of many other theorems by later Greeks such as Pythagoras and Plato. These two men (next to Thales) contributed the most to Greek geometry. Pythagoras discovered and proved many different theorems and ideas that contributed greatly to the development of geometry. Some of Pythagoras's proven discoveries included:
- All of the angles in a triangle add up to the sum of two right angles.
- The development and use of geometrical algebra.
- The theorem of Pythagoras. a^2 + b^2 = c^2
Pythagoras also did many studies with triangles and developing or editing shapes. His most famous discovery was the Pythagorean theorem (listed above). This theorem combined the sides of a right triangle, and this led to the development of irrational numbers by Pythagoras later on. Pythagoras discovered that the square root of 2 was an irrational number.
Plato, another great mind of Greece, did more than just develop theorems for geometry, he stressed that geometry was essential. Plato believed that everyone should be well educated in mathematics as well as geometry.
It is important to have some information about the organization that the strategic planning will cover. This section of the strategic plan gives a rapid review of the organization in order to understand the circumstances that the organization is performing in.
Study of Geometry gives students the tools to logical reasoning and deductive thinking to solve abstract equations. Geometry is an important mathematical concept to grasp as we use it in our life every day. Geometry is the study of shape- and there are shapes all around us. Examples of geometry in everyday life are- in sport, nature, games and architecture. The game Jenga involves geometry as it is important to keep the stack of tiles at a 90 degrees angle, otherwise the stack of tiles will fall over. Architects use geometry everyday- it is essential when designing buildings- shape, angles and area and perimeter are some of the geometry concepts architects
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
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.
This source provided a lot of background information on Euclid and his discoveries. This source gave details about the many geometrical theories of Euclid, as well as his practical geometrical uses. This source also explained how geometry helped Greece a long time ago, and how it is used by many people everyday.
...ed knowledge beyond ordinary people’s understandings. However, in some ways Aristotle did a better job than Plato. As a result, his ideas will continue to exist in this world for the generations yet to come.
If two sides of a triangle are congruent, then the angles opposite those sides are also congruent.
The Pythagoreans of 500 B.C Greece. believed that numbers were universal principles. Plato, who Platonism was named after, similarly argued that mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them. His belief was a century later than that of the Pythagoreans when established in 400 B.C. Euclid, who was acclaimed as the father of geometry, also believed that nature itself was the physical manifestation of mathematical laws. He did not believe that math came from nature, so it was a little different from the beliefs of other Platonists. So in a sense, he believed that the world was created because of math or to show mathematical
A student of Socrates, a major western civilization influence, and an amazing philosopher, Plato was his name and he was one of the most influential persons in history. Plato was born in Greece in 427 BC and grew up in a wealthy and noble family. He became a philosopher when his teacher, and another great philosopher of Greece, Socrates, was tried and executed in 399 BC. Plato wrote a lot about Socrates in his works of ancient Greece. Plato helped form classical education, and we would not have a good basis for education in America and western civilization. The first school of philosophy in Greece was the school that Plato established, in 387 BC, called the Academy. As a philosopher Plato had many theories. Some of which are still used in philosophy schools today. Plato traveled to Italy, Sicily, and Egypt spreading his philosophy over many different places. Plato’s school taught many of the ancient Greek thinkers, like Aristotle, who went on to teach Alexander the Great, who spread Plato’s and Aristotle’s teachings around the globe. Plato contributed to the thought process that is used in modern science today. Even though he was a mathematician he did not use math to figure things out in science, he used his knowledge of logic and he liked to reason with things and theories.
The Greeks were able to a lot of things with only a compass and a straight edge (although these were not their sole tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found means to construct parallel lines, to bisect angles, to construct various polygons, and to construct squares of equal or twice the area of a given polygon. However, three constructions that they failed to achieve with only those two tools were trisecting the angle, doubling the cube, and squaring the circle.
One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato's philosophy. Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music.
Physics began when man first started to study his surroundings. Early applications of physics include the invention of the wheel and of primitive weapons. The people who built Stone Henge had knowledge of physical mechanics in order to move the rocks and place them on top of each other. It was not until during the period of Greek culture that the first systematic treatment of physics started with the use of mechanics. Thales is often said to have been the first scientist, and the first Greek philosopher. He was an astronomer, merchant and mathematician, and after visiting Egypt he is said to have originated the science of deductive geometry. He also discovered theorems of elementary geometry and is said to have correctly predicted an eclipse of the sun. Many of his studies were in astronomy but he also observed static electricity. Phythogoras was a Greek philosopher. He discovered simple numerical ratios relating the musical tones of major consonances, to the length of the strings used in sounding them. The Pythagorean theorem was named after him, although this fundamental statements of deductive geometry was most likely first an idea from Egyptian methods of measurements. With the help of his followers he discovered that the earth was a sphere, but he did not believe it revolved around the sun.
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
Euclidean Geometry is a type of geometry created about 2400 years ago by the Greek mathematician, Euclid. Euclid studied points, lines and planes. The discoveries he made were organized into different theorems, postulates, definitions, and axioms. The ideas came up with were all written down in a set of books called Elements. Not only did Euclid state his ideas in Elements, but he proved them as well. Once he had one idea proven, Euclid would prove another idea that would have to be true based on what he had just discovered. Euclid was the first person to create this type of mathematical deduction. Out of all the mathematical discoveries Euclid made, one of the most famous would have to be the parallel postulate. The parallel postulate states that there is only one line that can be drawn through a point so that is parallel to another line not containing that point. To this day, the ideas that Euclid proposed are still relevant and taught in classrooms everywhere.
Plato is one of the most important people in the history of Philosophy. Throughout his life, he had made many contributions to the world of philosophy, but the most important contribution that he is most known for is his theory of the Ideas or Forms. Throughout his many works such as the Phaedo and Symposium, he presented his theory of Ideas by using both mythos and logos in his argument for support.