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.
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.
The Impact of Pythagoras, Socrates and Zeno of Elea
Pythagoras, a Greek, very influential philosopher back then but also nowadays. He was a mathematician and he founded the much known Pythagoras theory, but he is mostly known for his significance in numbers, he contributed in the developing of mathematics we know today. None of his works have survived unfortunately and back then it was not that easy to write and present your work, Pythagoras had to demonstrate his theorem using a stick in the sand. Another one of his works is the theory that all numbers are the key of understanding the world. Actually a theory goes around that some of his work was made up from his admirers after his death, but it hasn’t been confirmed yet. He was the first one to start making books and theories surrounding numbers and mathematics, even though Plato and Aristotle continued after him making a great name of themselves, but Pythagoras lived long before them.
Socrates was a Greek philosopher who influenced the modern philosophy in ways no other philosopher did. He was so known in Ancient Greece that people often talked about him in plays and was the target of many writers. Although he e anything of his own he impacted philosophy by his way of life and his way of doing things, some of his most notable studies are the study of the soul and the study of the universe. Xenophon is the work that talks about Socrates life and what kind of person he was, in the beginning it is said that he was a man of self mastery and had argumentative skills. Xenophon was like a formation of all his conversations that led his admirers to believe and live by the ideas Socrates gave. Socrates was sentenced to death by poisoning because he was hated in Athens and because according to sources he liked to humiliate people by making them appear as foolish. He was a critic of democracy which was loved by Athenians at the time, and he decided to go to trial rather than save himself, he gave a speech where he tried to defend himself and said the famous saying “the unexamined life is not worth living” which in his speech was refused and he died by getting poisoned.
Zeno of Elea was a Greek philosopher and mathematician who was widely known as the inventor of dialectic. He is known for his paradoxes that transformed mathematics. Zeno made use of three main premises the first one was that every unit has magnitude and he argued that if it is added or subtracted it does not increase or decrease, the second indicated that it is also infiently divisible and the third that it is invisible. He was arguing against one of his rivals which was Pythagoras. Pythagoras believed that numbers are extended units. Every mathematician at the time was having a hard time and has been contemplating the meaning of everything, but by doing that is how they got their results which we use freely today. Zeno’s statements of motion go by names given from Aristotle who studied and went over his paradoxes, today we know them as the Achilles paradox’s which proves that the slower mover will never be passed by the swifter race. It is proving that an object never really reaches the end. Zeno made some very controversial statements that to some his conclusions seem absurd and not very easily understood.
The philosophers I chose are all linked to each other in ways that I will now go over, Pythagoras and Zeno are both mathematicians who had a very hard time understanding their own thoughts, both of them not very known during their time but bloomed far after their death. Socrates was long after both of them and he is actually the most known and the most influential to philosophy since he had no impact in mathematics. Socrates gave philosophy some controversial ideas and quotes that truly shape modern philosophy nowadays, one of his greatest quotes is “One thing I know is that I know nothing” I understand this as he was very humble and accepting of the fact that he did actually not understand but yet he tried. Philosophy nowadays wouldn’t be the same without these three philosophers.
Mathematical and Philosophical Ideas of Pythagoras
Pythagoras was considered to be an influencer of many people and things around him. He was said to be interested in philosophy, astronomy, and music. He was greatly motivated by Pherekydes, Thales, and Anaximander. But one of the main achievements that he was known for is being the first real/famous mathematician.
Pythagoras was born approximately 569 BC at the island of Samos Greece. Historians are not really sure when he died but it is said that he died around 500 BC to 475 BC in Metapontum, Lucania, Italy. He was around 75 years old when he died. Pythagoras’s mother’s name was Parthenis, she was a native of Samos. She was from one of the islands most aristocratic families. His father’s name was Mnesarchus, he was a wealthy merchant of Phoenician heritage who had settled on Samos. Pythagoras father’s business was trade and traveled a lot. It was even said that he had two or three brothers. There also was a theory that he was married to a woman named Theano but others believed that it was one of his students that he was the teacher for (mathopenref.com). Some sources said he was treated like a God.
He was also the first to believe that the earth was round.
Pythagoras was most famous for a theory that he did the first proof on called the Pythagorean Theorem. This theory evolved in the world very quickly. This means that a right triangle the square of the hypotenuse is equal to a sum of two squares of other two sides. The equation was and still is a^2+b^2=c^2 Once this was put out to the world most people started to experiment with it. This is even still being used to this day in most math school settings like in geometry and algebra classes.
During his lifetime Pythagoras was also famous for some of the quotes that he wrote. One of the quotes was “There is geometry in the humming of the strings, there is music in the spacing of the spheres.” (Pythagoras) Another one was “Geometry is knowledge of the eternally existent.” (Pythagoras)“ He also had many other quotes that did not pertain to math but he was famous for them too. One was “Above the cloud with its shadow is the star with its light.” (Pythagoras)
He and his followers (which were called Pythagoreans) believed that everything can be a made into a number. They were also known for representing numbers by a series of equally spaced numbers by lines, triangles, and so on. This was a way that Pythagoras was able to see and change the shape with every number given. He and the Pythagoreans put the most important numbers into four basic groups. They are triangles, squares, and gnomons. Triangular numbers include any numbers that can form/be shaped into a triangle by a series of dots. An example would be numbers 3 and 10. There are also square numbers that are spaced into squares. An example would be the numbers 4 and 9. He was also known for something called Pythagorean triples. It is described as a group of positive numbers that follow the Pythagorean theorem formula which was a^2+b^2=c^2. One of them is (3,4,5).
Another way Pythagoras discovered math in music was by him to play the instrument called the lyre. When he played it, he realized that when he was playing a tune there are proportional numbers being involved. Some were 2:1 3:2 and 4:3. The ratio 2:1 represents a musical octave, 3:2 represents something called a musical fifth and 4:3 represents something called musical fourth. He even found out that his knowledge can be used in other instruments too. (mathopenref.com) With this Pythagoras and the Pythagoreans believed that music was the expression of harmony around the world. This was one of the many things that Pythagoras and the Pythagoreans were able to figure out the official connection with music and numbers. (Karamanides)
Pythagoras was the human being in his time period and was named a philosopher. It basically means a person who loves to search out for wisdom and enlightenment. He and the Pythagoreans (his followers) followed the teachings of the human soul and religion tied together with beliefs and some knowledge of science. Pythagoras was very intrigued with learning religious traditions in some countries and cultures. He believed the soul is made of three main parts. They were emotion, reason, and intelligence. He and the Pythagoreans believed that a person’s soul to be immortal and when dead it can be transmitted from one body to another (Karamanides 54-60). Pythagoras was also a man that believed in many things. One of them was “All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can be understood through mathematics.” Some other ones were “Numbers have personalities, characteristics, strengths and weaknesses.” and “Certain symbols have a mystical significance.” Another one was “All members of the society should observe strict loyalty and secrecy.” One more was “The world depends upon the interaction of opposites, such as the male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow (Mathopenref.com). The last belief he was known for was “The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of reincarnations called transmigration until it becomes pure (Mathopenref.com)
In Pythagoras’s lifetime he was also known as an astronomer. Pythagoras showed this belief of astronomy by saying the universe was made of ten spheres. These spheres were the sun, moon, earth and five other planets which were Mercury, Venus, Mars, Jupiter, and Saturn. He even said that there was something called the counter-earth. This was described as a global sphere which was never seen from the earth. Pythagoreans and Pythagoras then mixed astronomy with music. They said or believed that a planet that was moving faster made a very high pitch while it was orbiting the central-hearth. When a planet was moving slower it made less loud pitch of music. As shown above, Pythagoras was a man known for many things.
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- Karamanides, Dimitra. Pythagoras: Pioneering Mathematician and Musical Theorist of Ancient Greece. Rosen Central, 2006.
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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Pythagoras’ Biography and Mathematical Legacy
“Music is the harmonization of opposites, the unification of disparate things, and the conciliation of warring elements… Music is the basis of agreement among things in nature and of the best government in the universe. As a rule it assumes the guise of harmony in the universe, of lawful government in a state, and of a sensible way of life in the home. It brings together and unites.” – The Pythagoreans
Every school student will recognize his name as the originator of that theorem which offers many cheerful facts about the square on the hypotenuse. Many European philosophers will call him the father of philosophy. Many scientists will call him the father of science. To musicians, nonetheless, Pythagoras is the father of music. According to Johnston, it was a much told story that one day the young Pythagoras was passing a blacksmith’s shop and his ear was caught by the regular intervals of sounds from the anvil. When he discovered that the hammers were of different weights, it occured to him that the intervals might be related to those weights. Pythagoras was correct. Pythagorean philosophy maintained that all things are numbers. Based on the belief that numbers were the building blocks of everything, Pythagoras began linking numbers and music. Revolutionizing music, Pythagoras’ findings generated theorems and standards for musical scales, relationships, instruments, and creative formation. Musical scales became defined, and taught. Instrument makers began a precision approach to device construction. Composers developed new attitudes of composition that encompassed a foundation of numeric value in addition to melody. All three approaches were based on Pythagorean philosophy. Thus, Pythagoras’ relationship between numbers and music had a profound influence on future musical education, instrumentation, and composition.
The intrinsic discovery made by Pythagoras was the potential order to the chaos of music. Pythagoras began subdividing different intervals and pitches into distinct notes. Mathematically he divided intervals into wholes, thirds, and halves. “Four distinct musical ratios were discovered: the tone, its fourth, its fifth, and its octave.” (Johnston, 1989). From these ratios the Pythagorean scale was introduced. This scale revolutionized music. Pythagorean relationships of ratios held true for any initial pitch. This discovery, in turn, reformed musical education. “With the standardization of music, musical creativity could be recorded, taught, and reproduced.” (Rowell, 1983). Modern day finger exercises, such as the Hanons, are neither based on melody or creativity. They are simply based on the Pythagorean scale, and are executed from various initial pitches. Creating a foundation for musical representation, works became recordable. From the Pythagorean scale and simple mathematical calculations, different scales or modes were developed. “The Dorian, Lydian, Locrian, and Ecclesiastical modes were all developed from the foundation of Pythagoras.” (Johnston, 1989). “The basic foundations of musical education are based on the various modes of scalar relationships.” (Ferrara, 1991). Pythagoras’ discoveries created a starting point for structured music. From this, diverse educational schemes were created upon basic themes. Pythagoras and his mathematics created the foundation for musical education as it is now known. According to Rowell, Pythagoras began his experiments demonstrating the tones of bells of different sizes. “Bells of variant size produce different harmonic ratios.” (Ferrara, 1991). Analyzing the different ratios, Pythagoras began defining different musical pitches based on bell diameter, and density. “Based on Pythagorean harmonic relationships, and Pythagorean geometry, bell-makers began constructing bells with the principal pitch prime tone, and hum tones consisting of a fourth, a fifth, and the octave.” (Johnston, 1989).
Ironically or coincidentally, these tones were all members of the Pythagorean scale. In addition, Pythagoras initiated comparable experimentation with pipes of different lengths. Through this method of study he unearthed two astonishing inferences. When pipes of different lengths were hammered, they emitted different pitches, and when air was passed through these pipes respectively, alike results were attained. This sparked a revolution in the construction of melodic percussive instruments, as well as the wind instruments. Similarly, Pythagoras studied strings of different thickness stretched over altered lengths, and found another instance of numeric, musical correspondence. He discovered the initial length generated the strings primary tone, while dissecting the string in half yielded an octave, thirds produced a fifth, quarters produced a fourth, and fifths produced a third. “The circumstances around Pythagoras’ discovery in relation to strings and their resonance is astounding, and these catalyzed the production of stringed instruments.” (Benade, 1976). In a way, music is lucky that Pythagoras’ attitude to experimentation was as it was. His insight was indeed correct, and the realms of instrumentation would never be the same again.
Furthermore, many composers adapted a mathematical model for music. According to Rowell, Schillinger, a famous composer, and musical teacher of Gershwin, suggested an array of procedures for deriving new scales, rhythms, and structures by applying various mathematical transformations and permutations. His approach was enormously popular, and widely respected. “The influence comes from a Pythagoreanism. Wherever this system has been successfully used, it has been by composers who were already well trained enough to distinguish the musical results.”
In 1804, Ludwig van Beethoven began growing deaf. He had begun composing at age seven and would compose another twenty-five years after his impairment took full effect. Creating music in a state of inaudibility, Beethoven had to rely on the relationships between pitches to produce his music. “Composers, such as Beethoven, could rely on the structured musical relationships that instructed their creativity.” (Ferrara, 1991). Without Pythagorean musical structure, Beethoven could not have created many of his astounding compositions, and would have failed to establish himself as one of the two greatest musicians of all time. Speaking of the greatest musicians of all time, perhaps another name comes to mind, Wolfgang Amadeus Mozart. “Mozart is clearly the greatest musician who ever lived.” (Ferrara, 1991). Mozart composed within the arena of his own mind. When he spoke to musicians in his orchestra, he spoke in relationship terms of thirds, fourths and fifths, and many others. Within deep analysis of Mozart’s music, musical scholars have discovered distinct similarities within his composition technique.
According to Rowell, initially within a Mozart composition, Mozart introduces a primary melodic theme. He then reproduces that melody in a different pitch using mathematical transposition. After this, a second melodic theme is created. Returning to the initial theme, Mozart spirals the melody through a number of pitch changes, and returns the listener to the original pitch that began their journey. “Mozart’s comprehension of mathematics and melody is inequitable to other composers. This is clearly evident in one of his most famous works, his symphony number forty in G-minor” (Ferrara, 1991). Without the structure of musical relationship these aforementioned musicians could not have achieved their musical aspirations. Pythagorean theories created the basis for their musical endeavours. Mathematical music would not have been produced without these theories. Without audibility, consequently, music has no value, unless the relationship between written and performed music is so clearly defined, that it achieves a new sense of mental audibility to the Pythagorean skilled listener..
As clearly stated above, Pythagoras’ correlation between music and numbers influenced musical members in every aspect of musical creation. His conceptualization and experimentation molded modern musical practices, instruments, and music itself into what it is today. What Pathagoras found so wonderful was that his elegant, abstract train of thought produced something that people everywhere already knew to be aesthetically pleasing. Ultimately music is how our brains intrepret the arithmetic, or the sounds, or the nerve impulses and how our interpretation matches what the performers, instrument makers, and composers thought they were doing during their respective creation. Pythagoras simply mathematized a foundation for these occurances. “He had discovered a connection between arithmetic and aesthetics, between the natural world and the human soul. Perhaps the same unifying principle could be applied elsewhere; and where better to try then with the puzzle of the heavens themselves.” (Ferrara, 1983).
How Pythagoras of Samos Viewed Music
Aldous Huxley, one of the most preeminent intellectuals of his time, stated, “after silence, that which comes nearest to expressing the inexpressible is music.”(goodreads) Never a more verbally inexpressible art many of the world’s greatest minds have been drawn to this subject that bridges body and soul. Einstein stated, “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.” The logical reasoning of the head does not tend to understand the illogical reasoning of the heart. For many intellectuals, music is the bridge that leads to that understanding. Pythagoras saw music as the foundations of life as we know it. Every leaf, every planet, every person has a melody and rhythm of their own.
The life and mathematical achievements of Pythagoras of Samos, born around 570 BCE and died around 490 BCE, is a mystery to the mathematical world. Due to there being no surviving documents of Pythagoras it is unknown as to what the man actually did himself and what his followers did in his name. Though it is uncertain, still many innovations have been connected to the mathematician. Specifically, Pythagoras made music related mathematical achievements which are exemplified through his life, the music of the spheres, and the musical octave.
Said to be “the first pure mathematician”, Pythagoras of Samos is very important to the mathematics that we know and love today. Sadly, though he has been given this title, as stated above very little is known about his achievements in the field. There is nothing of Pythagoras’s writings that has survived to this day. This could be in part due to the secrecy of this influential man. “The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. “ (O’Connor) Pythagoras spent some time in Egypt and it’s said that it was during this time that many of his beliefs were molded. “The accounts of Pythagoras’s time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. It is not difficult to relate many of Pythagoras’s beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. (history)
To speak nothing on the fact that some of what is said of Pythagoras is pure myth would be a failure in giving an accurate summary of his life. The myths are said to be possible because of a trend started in the first centuries BCE in which Pythagoras was talked of as if he was a “semi-divine“ figure rather than just an ordinary man with an extraordinary mind. Writings of even plato and Aristotle began to be attributed to Pythagoras in an attempt to support this dissertation. “The Pythagorean question, then, is how to get behind this false glorification of Pythagoras in order to determine what the historical Pythagoras actually thought and did.“ (plato)
Pythagoras’s mother,Pythais, was a native of Samos. His father was named Mnersarchus and it is said that he brought corn to Samos as a traveling merchant from Tyre during a time of great famine. This gained him citizenship from the people as a show of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father.” (history) “There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras. The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In  it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old.Thales was an old man by this time and had little practical knowledge to teach Pythagoras. He better served as a mentor in the time they shared, creating a strong impression on Pythagoras and inspiring him to pursue deeper knowledge of mathematics and astronomy in Egypt. Thales’s pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras’s own views.” (O’Connor)
The path in which Pythagoras came to his “Musica Mundana”, Music of the Spheres, is centered around a story unknown as to if it is true or false. This should be of little surprise given what we know about Pythagoras. It is said that the man, as he walked by, took note of the sounds of blacksmiths’ hammers as they beat down on an object. The hammers were said to have different sounds based on their size. (Aboutscotland) This lead Pythagoras to go home and tie weights the same as each hammer to strings and pluck them to see what sound they made. He was said to have been an expert lyre player and as such would have had a working knowledge of strings based on that alone.(philclub.org)
This string theory is the beginning of both Musica Mundana as well as the octave as musicians know it today. Changing proportions on the strings changes frequencies. It was Pythagoras who noticed that the multiples of the same frequencies could be considered the same note.
Therefore, once a frequency is known, either halving or doubling the frequency will give the same note name below and above respectively for that specific note value; these are called octaves. As an example, an A1 on a piano has a frequency of 55.00Hz while A3 has a frequency of 220.00Hz. Since A1 is doubled to create A2 and A2 is doubled to create A3, the equation 2×2=4 is created. 55.00 multiplied by 4 is 220.00. To further the example, to get back to A2, A3 would need to be halved. Half of 220.00 Hz is 110.00 Hz. Upon looking at figure 1, it can be seen that A2 is in fact, 110.00 Hz. “If a taut string is merely touched at the center, so that the ratio of subdivided intervals is exactly 1:1, the string will emit a note that is an octave higher than the fundamental of the string. The note of the stopped string is also an octave higher. This stopped note is in a ratio of 1:2 in wavelength to open string. Thus its frequency will be twice as high, because it’s wavelength is twice as short. Here we see that Pythagoras was developing the beginnings of a wave theory.” (philclubcle) Different proportions will show the other notes within the modern harmonic progression. “By dividing the string into various other lengths, intervals of the fourth and fifth were produced, and so on. Pythagorasn and his followers conceived of the universe a vast lyre in which each planet, vibrating at a specific pitch, in relationships similar to the stopping of the monochord’s string, harmonized with other heavenly bodies to create a “music of the spheres.” (Richards)
Lamda, a “triangular figure of numbers, is the Tetrad of the Pythagoreans… It is a set of numbers whose relationships with each other seemed to summarize all the interdependent harmonies within the universe of space and time.” (aboutscotland) This set of numbers is an summary for everything including the planets. Pythagoras taught that each of the seven planets produced by its orbit a particular note according to its distance from the still center which was the earth.” (aboutscotland) Figure 3 shows the information on the notes for each of the planets in our solar system.