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.
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Ancient Greece's philosophers and mathematicians have made contributions to western civilizations. Socrates believed that a person must ask questions and seek to understand the world around them. Aristotle, another famous philosopher, is known for believing that if people study the origin of life, they will understand it more. Reasoning is what makes human beings unique. Hippocrates was a mathematician and a doctor. He created the Hippocratic oath. The oath states that Hippocrates will treat his patient to the best of his abilities that he will refuse to give deadly medicine. This oath is still used by doctors today. Another Greek mathematician was Euclid. His ideas were the starting point of geometry, which is still studied around the world today.
Euclid’s Elements are predominantly the most fundamental concepts of mathematics, but his perspective on geometry was the model for over two millennia. He is believed by many to be the leading mathematics teacher of all time. However, little is known about his life outside of mathematics, or even when he was born or when he died. According to a passage written by Proclus, Euclid probably lived after Ptolemy and the pupils of Plato, but came before Archimedes and Eratosthenes. This places his existence sometime around 300 B.C. Euclid is most famous for having set the guidelines for geometry and arithmetic written in Euclid’s Elements, a series of thirteen books in which Euclid states definitions, postulates, and theorems for mathematical concepts that are still used today. What is most remarkable about the Elements is the simple, rational, and very logical structure in which Euclid presents the accumulated geometrical knowledge from the past several centuries of Greek mathematicians. The manner in which the propositions have been derived is considered to be the prime model of the axiomatic method. (Hartshorne 296).
One of the most well known contributors to math from Greece would be Archimedes. He
In Chapter 2 of Journey Through Genius, titled “Euclid’s Proof of the Pythagorean Theorem,” the author, William Dunham begins by introducing the Greek contributions to mathematics. The first figure introduced, Plato, brought enthusiasm to the subject. He was not an actual mathematician; he was a philosopher. His main contribution to math was establishing the Academy, a center devoted to “learning and contemplation for talented scholars.” The Academy was mainly focused on mathematics and produced talented scholars, such as Eudoxus. Eudoxus was another important figure in math, as he created the theory of proportion and the method of exhaustion. When Alexander the Great set out to “conquer the world,” he set up a city in Egypt called Alexandria
Greek mathematics began during the 6th century B.C.E. However, we do not know much about why people did mathematics during that time. There are no records of mathematicians’ thoughts about their work, their goals, or their methods (Hodgkin, 40). Regardless of the motivation for pursuing mathematical astronomy, we see some impressive mathematical books written by Hippocrates, Plato, Eudoxus, Euclid, Archimedes, Apollonius, Hipparchus, Heron and Ptolemy. I will argue that Ptolemy was the most integral part of the history of Greek astronomy.
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.
For the Greeks philosophy wasn’t restricted to the abstract it was also their natural science. In this way their philosophers were also their scientist. Questions such as what is the nature of reality and how do we know what is real are two of the fundamental questions they sought to answer. Pythagoras and Plato were two of the natural philosophers who sought to explain these universal principles. Pythagoras felt that all things could be explained and represented by mathematical formulae. Plato, Socrate’s most important disciple, believed that the world was divided into two realms, the visible and the intelligible. Part of the world, the visible, we could grasp with the five senses, but the intelligible we could only grasp with our minds. In their own way they both sought to explain the nature of reality and how we could know what is real.
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 Pythagorean Theorem is the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is more commonly known as x2+y2=z2. What is important is the triangle must have a 90 degree angle to make this theorem true. Within this are special values that are called Pythagorean Triples. The Pythagorean Theorem is only useful if one follows the rules that come with it.
While he was born thousands of years ago, Euclid has made a mark in mathematics that has stood the test of time and continues to shape modern sciences. He discovered classic geometry and wrote many book and papers on mathematics that proved useful to the great thinkers of the time. Having drafted a lot of his finding in his book The Elements, he has made a framework for mathematicians and is, even 23 centuries later, revered as valued information. His years of studying have made him a loved and respected individual. Euclid went from just another well off individual in the times of the great era of Greece to a man still remembered today as “The Father of Geometry”.
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.
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).
Euclid, also known as Euclid of Alexandria, lived from 323-283 BC. He was a famous Greek mathematician, often referred to as the ‘Father of Geometry”. The dates of his existence were so long ago that the date and place of Euclid’s birth and the date and circumstances of his death are unknown, and only is roughly estimated in proximity to figures mentioned in references around the world. Alexandria was a broad teacher that taught lessons across the world. He taught at Alexandria in Egypt. Euclid’s most well-known work is his treatise on geometry: The Elements. His Elements is one of the most influential works in the history of mathematics, serving as the source textbook for teaching mathematics on different grade levels. His geometry work was used especially from the time of publication until the late 19th and early 20th century Euclid reasoned the principles of what is now called Euclidean geometry, which came from a small set of axioms on the Elements. Euclid was also famous for writing books using the topic on perspective, conic sections, spherical geometry, number theory, and rigor.
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.
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.