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# Pythagoras

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## Ancient Greek Mathematicians: Thales, Pythagoras, Euclid, and Archimedes

“Geometry is knowledge of the eternally existent,” (“Sacred Mathematics”). This quotation by Plato, an Ancient Greek philosopher, demonstrates the importance of geometry to the foundations of the universe. Geometry encompasses every aspect of life including architecture, physics, and biology. Teachers around the globe instruct the basics of geometry to teen-aged students every day, yet these self-evident ideas were not always simple. It took the collaboration of many great minds to formulate the mathematical conclusions so easily comprehensible today. Ancient Greece’s thriving civilization allowed great thinkers such as Thales, Pythagoras, Euclid, and Archimedes to flourish through discovery and innovation. Because of the considerable time period, these mathematicians belong to one of two categories: the early mathematicians (700-400 BCE) and the later mathematicians (300-200 BCE). Thales and Pythagoras are early mathematicians, while Euclid and Archimedes are later mathematicians. Their discoveries provided a better understanding of geometry and developed the principle understandings of the world around us, thus providing invaluable contributions to the field of mathematics, especially in geometry.

## Thales: The Father of Greek Mathematics

One of the earliest great Greek mathematicians was Thales. Thales (624-560 BCE) was born in Miletus, but resided in Egypt for a portion of his life. He returned to Miletus later in his life and began to introduce and shape his knowledge of astronomy and mathematics to Greece (Allman 7). As an astronomer, he was infamous for accurately predicting the solar eclipse on May 28, 585 BC. But, evidence points to this prediction being a fluke as astronomy at the time was not advanced enough to make such a prediction (Symonds and Scott 2).

Mathematically, however, his contributions are more reputable. Historians believe that Thales introduced the concept of geometry to Greece (“Thales”). Through his use of logical reasoning and his view of geometrical figures as mere ideas rather than physical representations, Thales drew five conclusions about geometry. In circular geometry, Thales proved the diameter of a circle perfectly bisects the circle, and that an angle inscribed in a semicircle is invariably a right angle. In trigonometry (the geometry of triangles) he discovered that an isosceles triangle’s base angles are equal. Today, architects still rely on this principle to ensure that steeples and spires on buildings are level. He also proved that triangles with two congruent angles and one congruent side with each other are, in themselves, congruent, as displayed. Later, artists used this proof in paintings to ensure symmetry, particularly in modern works. Lastly, he proved that when two straight lines intersect, the opposite angles between the two lines equal each other (Symonds and Scott 1), which is crucial to predicting trajectory in physics.

His version of geometry was abstract for the time period, as “Thales insisted that geometric statements be established by deductive reasoning rather than by trial and error” (Greenberg 6). He focused on the relationships of the parts of a figure to determine the properties of the remaining pieces of the figure (Allman 7). Through his discoveries, Thales influenced his successors and aided in their discoveries. But, he also applied them practically to Grecian life. The theorems he formed on congruent triangles and their corresponding parts and angles allowed him to more accurately calculate distances, which ultimately aided in sea navigation (Wilson 80), crucial due to this being their main mode of transportation. Thales’ ideas also founded the geometry of lines, “which has ever since been the principal part of geometry,” (Allman 15). Through his development of this principal part of geometry and his exposure of these ideas to the Greeks, Thales greatly impacted the overall development of mathematics. Historians acknowledged these contributions by naming Thales as one of the Seven Wise Men of Greece (“Thales”).

## Pythagoras: The Father of Trigonometry

Living from 569-500 BCE, Pythagoras, too, found an interest in mathematics and astronomy as he studied under one of Thales’ pupils, Alzimandar. Through his years of research and study of mathematics, Pythagoras attracted a community of followers in his home of Crotona (Wilson 80). Known as the Pythagoreans, scholars credit them with discovering the sum of the angles of a triangle equalling two right angles, or 180 degrees, and the existence of irrational numbers. Another notable accomplishment is the construction of the five regular solids: the tetrahedron, the hexahedron (cube), the octahedron, the dodecahedron, and the icosahedron (Polyhedron). Later, scientists found these solids to represent the atomic shapes of compounds. Today, students and educators alike most recognize Pythagoras for the Pythagorean theorem, in which “the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the other two sides” (Symonds and Scott 3), or a^2 + b^2 = c^2. This theorem developed the basic principle of trigonometry, which is the basis of physics.

Eventually, however, the Pythagoreans particularly focused on abstract rather than concrete problems. (Symonds and Scott 3). Rather than focusing on measurable, concrete quantities as numbers, the “Pythagorean worldview was based on the idea that the universe consists of an infinite number of negligibly small indivisible particles” (Naziev 175). This group believed that the objects around them (water, rocks, materials) were all constructed of microscopic, single units, later discovered to be atoms. It is through this assertion that Pythagoras coined his slogan, “All is number.” Through this sentiment, he implies that everything in the universe can be explained, organized, and predicted using numbers and mathematics. (M. B. 47).

## Euclid: The Father of Geometry

Euclid, the first well-known mathematician from Alexandria, lived from 325-265 BCE. (Wilson 96). Euclid attended a Platonic school, where he found his passion for mathematics and logic (Greenberg 7). He is most well known for his collection of his plane and solid geometry studies: his book Elements. Influenced by Thales’ geometrical beliefs, Euclid wrote his Elements to serve as an example of deductive reasoning in practice, starting with “initial axioms and deduc[ing] new propositions in a logical and systematic order.” (Wilson 96). Consisting of thirteen books covering topics from arithmetic, plane and solid geometry, and number theory, its groundbreaking content and overall influence catapulted this work to become one of the greatest textbooks in history, being the second most sold book only to the Bible. And, because of the success of this title, experts recorded Euclid as the most widely read author in history (Greenberg 7), in addition to one of the greatest mathematicians of all time (Symonds and Scott 4).

The first four volumes of Elements focus on the Pythagoreans and some of their discoveries (Greenberg 7). The fifth volume is said to be the “finest discovery of Greek mathematics” as it explains geometry as dependent on recognizing proportions, and the sixth volume applies these proportions to plane geometry. Volumes seven through nine focus on number theory, while volume ten deals with irrational numbers. Lastly, the eleventh through thirteenth volumes focus on three-dimensional geometry (Symonds and Scott 4). Some examples of the content in these volumes include the five postulates in volume one. Euclid writes,

Let the following be postulated: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines … meet on that side on which are the angles less than the two right angles. (Euclid 2) Through these postulates, Euclid focuses his propositions, through which he makes discoveries including measurements of angles in constructions, bisections, and proportional lengths (Euclid 2-36).

Euclid’s discoveries span across multiple principles of geometry. These discoveries serve crucial roles in construction and architecture, helping build and produce buildings with structural integrity while aiding the workers in accurately estimating the amount of material needed to complete a project. Not only do these aid in construction, but his discoveries apply greatly to engineering and physics. Through his focus on angles, he created the basis for future scientists to predict trajectory and to aid athletes in optimizing their performance.

## Archimedes: The Father of Mathematics

Born in 287 BC, Archimedes of Syracuse on the island Sicily studied mathematics and, because of his discoveries, scholars consider him to be one of the top-ranking mathematicians of all time (Symonds and Scott 4-5). Archimedes continued some of the work of the Pythagoreans and Euclid as he recorded the thirteen semi-regular solids. Scientists later discovered that these solids serve as depictions of some crystalline structures. Influenced by Euclid’s Elements, he also found the surface areas and volumes of spheres and cylinders used to determine the amount of a substance in a can or the amount of air needed to expand a balloon to a specific size (Wilson 96). Following this discovery, Archimedes detailed his ability to find these properties, saying, “These properties were all along naturally inherent already in their figures referred to, but they were unknown to those who were before our time engaged in the study of geometry, because none of them realized that there exists symmetry between these figures” (Dijksterhuis 142), just as Euclid determined in the fifth volume of Elements. Archimedes found these surface areas and volumes by calculating the ratios for these solids to circles and to each other. For instance, the surface area of a sphere is 4πr2 units while the area of a circle is πr2 units, leaving the surface area of a sphere and the area of a circle in a perfect 4:1 ratio. Similarly, the volume of a cylinder is πr2h units while the area of a circle is πr2 units, meaning these two properties are also directly proportional.

While recognizing the proportionality of these components, Archimedes also decided to focus on “the ratio of the circumference of a circle to its diameter”: pi. By drawing polygons with numerous sides inscribed in a circle and calculating the perimeter of said polygons, he was able to more accurately compute the value of pi. He found pi to lie between 3 10/71 and 3 1/7, the most precise prediction for his time period. (Symonds and Scott 5). This estimation for pi allowed a more accurate calculation of the area of a circle and volumes of spheres and cylinders, which can be applied to the calculation of sound waves to better understand pitch for music.

In computing these components of circular objects, Archimedes actually perfected a method of integration (Symonds and Scott 5), commonplace in calculus which, as a section of mathematics, wasn’t invented until much later. Archimedes was way ahead of his time in discovering this method, and this allowed himself and future mathematicians to calculate areas and volumes for various shapes.

## Discussion

The Ancient Greek mathematicians contributed to mathematics more than they could have predicted. Many of these people found interest in the field through their studies of prior mathematicians, and capitalized on prior discoveries to draw their own conclusions. This group of people were some of the first to study principles that were abstract and did not require physical tests to prove; rather, they relied on deductive reasoning to develop their theorems. This practice set the precedent for all future scientific and mathematical discoveries. The Ancient Greek mathematicians influenced not only the mathematics of their times or the mathematics of the future, but the overall process of all further scientific discoveries and experiments, thus proving to be invaluable assets to both the field of mathematics and scientific thought as a whole.

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## Pythagorean Theorem: the Heart of Mathematics

Have you ever wondered what the Pythagorean theorem is? Where it came from? Who came up with the idea of it? I’m going to start by telling you the history of the Pythagorean theorem. In this part of the essay you will learn the timeline of the Pythagorean theorem and its Importance in history. Although even being around for over 4000 years its importance and uses are almost limitless. Its effects are everywhere and affect everyone and everything. The affects are found in living and nonliving things. Although the effects seem like they are hidden they are right in front of our eyes. It’s simple to understand once you learn about them. Even the math is simple and can be very versatile and can be used with many other formulas to improve its solving capabilities. Not only is the math and simple but the history of this theorem is interesting. From this theorem being fo und by Babylonians to being discovered by Pythagoras to being passed on to his followers. From his followers it has new been proven over 300 times and still proofs are being found. More and more this theorem evolves and gets more and more complex in the manner used. None of this would make scenes though if not explained and an explanation is due. What is the history of this theorem? How does the mathematics work? How is it used today? Not only can those questions be answered but they can be explained, proven and shown. Even now there is so much unexplained about the Pythagorean theorem. This essay will only provide a skin to the whole system that is this theorem. This theorem is easily explained but understanding it and the history about it is very difficult. The origins of this theorem are unknown and most likely will never be found but as time goes on our understanding and uses for this theorem will be way greater and farther complicated.

The Pythagorean theorem is one of the oldest mathematical theorems known to men. It goes way back in 1900-1600 BCE but was looked more into and resolved Pythagoras. Pythagoras was a mathematician and a philosopher. He founded a brotherhood that had contributed to development of mathematics. Even inspiring and influencing Plato and Aristotle with his formulated principles and philosophies. Even with all the developmental stuff he has done for mathematics none of his true writings and teachings have survive through word of mouth of his followers. As many know word of mouth travels and changes due to interpretation. Although traveling and changing from his followers’ interpretations the Pythagorean theorem has also stayed the same without fail. Use of the theorem can be seen everywhere in history. It is used I all architecture through time.it has been proved many times throughout history and has been estimated to have 367 different proofs. Not only has this theorem been used for scales and buildings it is used for supports in bridges. A more ancient type of architecture that used this theorem for calculation for supports is aqueducts. Every support pillar has three support triangles to help keep the weight from crumbling the pillars. Triangles are important in architecture because the equally distribute pressure and weight throughout the triangle. This was useful because of the lack of materials and supplies available. It allowed less materials to be used and still have buildings be sturdier while standing under all its own weight.

Even before the Pythagorean theorem triangles were being used in buildings for supports but it wasn’t until Pythagoras discovery that buildings became more and more complex. Look at the colosseum. The colosseum is 1949 years old and is still standing strong. Even buildings such as the leaning tower of Pisa still stands even after its awkward and unusual state of being. For over 4000 years its seen in architecture and teaching by scholars in all ages of time. Even as children we learn about it.at your ages we learn about the uses and traits of right triangles. These weird traits were stating to be questioned by the Chinese in the 1st that were not yet realized could be solved by the theorem. One of these problems being having the ability to find the length of a side of a right triangle only being given a combination of either two angles or two side or both. Today we have just interpreted that into the theorem but that wasn’t the true purpose of the theorem. Seeing as the theorem was discovered mainly by seeing that triangles and right triangles can be fit into any regular shape. Most of these being squares and rectangles. This can also work with some irregular shapes. this can be seen in the game tangrams. There isn’t much left to say or explain about the history of the theorem. Most ancient examples can only be seen in buildings and seen on tablets that were used only as examples. Most of the knowledge died when Pythagoras and his followers and following philosophers died. Even though he wasn’t the one who created it he perfected it and taught the world of the masteries of this amazing and versatile theorem. It is only time till the theorem is perfected and more is discovered because Mathematics has evolved and changed and become more advance as time goes on. Even as time continues, and all these advancements happen this theorem will always be helpful.

What about how this theorem works. How does a theorem provide this much information and when can you use it? What are its limits and how does it benefit the user? In this part of the essay you will learn how the theorem works mathematically and physically makes sense. With most math it is hard to make it physically work but this theorem is possible to be created outside of writing.

When it comes to the Pythagorean theorem it is hard to tell when you use it and when you don’t. It’s as easy as ABC. a2 +b2=c2 is the equation. This equation only works with right triangles and can be used to find any of the sides of the triangle. Depending on the side you are trying to find the equation can change from addition to subtraction. It starts with a, A is usually the smallest or the side that is flat to the bottom. B is the second longest or the side that goes up. C is the hypotenuse or the longest side and is usually the side you will be looking for. If you know what two of the sides of the triangle are you can always find the third. When you only have one side of the triangle the formula can still be used but there can be a very of answers for the other two sides unless you use another formula that uses the right angle or any of the other angles to help you calculate the lengths. This isn’t part of the Pythagorean theorem, but it can be used to help find all the side of the triangle. This is because the Pythagorean theorem isn’t perfect.it isn’t able to find the hypotenuse without two sides already known. With help from other formulas and equations this theorem can cover a vast majority of materials. Through these vast amounts of materials, you can physically show the how this equation works. First you make squares of all the sides. Taking the area of both the smaller squares and combining them will equal the area of the bigger square. This is the most accurate way for finding unknown sides of a triangle. This can be used to help find a rough estimate of the unknown sides of any type of triangle, but it is always correct when used for right triangles.

Although this is extremely accurate and is always correct when used in the right way it is very faulty. First it only works with right triangles. If used on equal lateral triangles for example c would equal twice the amount that it really would. When it comes to the type of triangles it only works with one specific type of triangle. Although it can be used to estimate other types of triangles it won’t be as accurate as right triangles. There are little to no disadvantages to this theorem. If used in the correct way, it will almost end up right and is easy to learn and you can learn other equations to further the uses of this theorem. Every day we see how the Pythagorean theorem is used. It is everywhere around us in both natural and manmade objects. Not only is it there but we also use the effects of it every day and every second of our lives.

In architecture triangles are the most important part. Triangles are used for equal weight distribution. They are usually right triangle and put in between wall beams and used in roofs they are used as supports under a bridge and even the wires on top of a bridge. When it comes to supports and weight triangles are the only shape that can successfully at holding buildings together. Occasionally, you’ll see a board holding up a fence. Imagine the ground is side a and the fence is side b what is side c. Side c is the board touching the ground and the fence. Side c is the hypotenuse. If you knew the height of the fence and the length of the ground from the fence to the end of the hypotenuse, you would be able to calculate a highly accurate estimate of how long the hypotenuse is. This is just one common example of how people use the Pythagorean theorem. This theorem is a very common practice in our lives. It’s used when you want your picture frames to stand up. This theorem is used in our daily lives as well as our jobs. It is everywhere you just must look for it or know what you’re looking for.

In natural objects the Pythagorean theorem is also used. Unlike in manmade objects it’s hard for us to see it. You would have to go out of your way to find them but there are still common examples. In gemstones triangles are natural forming because of the way they are created to support weight. this can also be seen in the patterns that cut diamonds are in today. even in the smallest of jewelry and toys you can see triangles at work. In common toys such as Legos you can see the supports are right triangles and even that they are extremely durables and withstand the pressure of children and adults’ feet.

Even in plain sight people would still question how that is the Pythagorean theorem is used every day in common life. It isn’t directly used unless being taught or doing homework. In our heads we don’t usually do this math all the time. We use the effects of this theorem. This theorem affects our daily life from simply objects like toys and furniture to more complex objects like buildings and bridges. It’s not that we use the theorem is that we are using the after effects of the theorem and the theorem was used to create items so we can use them. It’s used to create items to be more stable and durable.jobs such as an architect carpenter metal worker or welder use this theorem during work and use it constantly for their job but it isn’t used in many other jobs.

The Pythagorean theorem serves a big purpose on our day to day living. There are many real worlds uses for it. It is used in our day to day lives even without knowing it. The Pythagorean theorem is a big component in our lives, and I have learned to appreciate it a little more after writing this paper. Not only have I learned how to use it, but once it is learned and you know how it is used you are able to see its affects everywhere.

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