Category: Pythagoras

Pythagoras made a lot of mathematical and mystical contribution to the modern numerology. But after his death, people’s interest in mathematical mysticism and all his teachings on numbers waned. Most of his teachings were now restricted to secret use. Sometime after his death however, a group of neo-Pythagoreans emerged and brought up his teachings again. But eventually the non-mathematical works and theories Pythagoras faded away. In addition to the death of Pythagoras, the ruling of the first council of Nicaea in the Roman Empire also contributed to the decline of numerology. After the council, alterations and antithesis of the beliefs of the state church was considered a civil violation all over the Empire. It was clear that the Church authority in the Empire didn’t fancy numerology and its principles; it was further classified under the unapproved beliefs of the time alongside magic and other types of divination. Regardless of the religious attacks and disagreement, the numbers of numerology still has a high level of spiritual significance. Some philosophers came up with some spiritual numbers like the Jesus number plus the fact that some Greek orthodox gatherings still find numerology useful to them. Also, in the course of the disapproval from the church, several numerology faithful argued that the principles of numerology can also be clearly seen in the bible where numbers like 5, 7 and 3 have a lot of spiritual significance and meaning. For instance, God created the world in three days, Jesus is said to have been dead for three days before he resurrected, the numbers 7 and 3 was highly associated with the activities of God in the bible more than any other number.

Other philosophers like Cornelius Agrippa wrote on numerology during Pythagoras time and even after his death. Cornelius Agrippa in his work occult philosophy (1533) provided parts of the ground in which western numerology is practiced now. He strongly was of the opinion that numbers have inherent powers in them and can be used for both physical and spiritual purposes. For example he identified some numbers as tools used for healing, casting out evil spirits or summoning good ones. For example, he pointed out that the number five has the power and energy to cast demons and to serve as anti-poison.

Apart from the studies of Pythagoras, the origin of numerology has also been attributed to some alchemical theories which have closely related principle and beliefs with that of numerology. This is evidently seen in the work of Arab alchemist Jabir Ibn Hayyan whose experiments and researches were basically founded on principles and assumptions of numerology. They were based on the names of substances in the Arabic language. Another notable work that highlighted numerology before the modern numerologists was the literary discourse of Thomas Browne: the Garden of Cyrus. In the work, he established the sacredness of the number five and its corresponding Quincunx pattern which can be seen clearly in nature and designs, mostly in botany.

After the contributions of Pythagoras and his followers, other modern numerologists began to write and most of them were building on the works of Pythagoras. One of the most notable works among others was that of Mrs. Dow Balliett who published lots of books and articles on vibration, music and colors. She particularly built from and integrated her works with the Pythagoras teachings and principles. Her work “The romance in your name” published by her student Juno Jordan in 1965 served as a major foundation of the numerology system used in modern or western numerology now. Balliett and other modern numerologists, posits that every number has a vibration unique to it. Not just numbers in fact, everything has its own vibration; colors, foods, colors, people etc. these unique vibrations play a very important role in the lives of every individual. And if anybody will have a good and productive life, they have to make sure that their vibrations are in perfect harmony with the other vibrations around their environment. This particular concept of numerology is also seen in other astrological and new age practices asides numerology and is described to be as a result of the movement of subatomic particles. Although this is a main concept of numerology, there is not enough scientific research that identifies, measures these vibrations and the effect it has on people and the environment generally. Some other numerologists have often established a kind of connection between these vibrations and the sounds that comes from the planets and the sun while orbiting the earth according to Pythagoras. According to Pythagoras, all the planets are enclosed in a kind of transparent physical sphere and that there is a corresponding musical ratio with the distance between the planets. Some of Pythagoras assertions relating to vibrations as well as that of the sun orbiting the earth have been nullified by science.

Modern numerologists have also made a number of modifications on some of the principles of Pythagoras. For example, they attach numbers to people and also attach intangible concepts to numbers. Most of them believe that the numbers one to nine have corresponding unique properties due the different vibrations natural to them. The properties and connotations of numbers are derived from the peculiar ways different cultures use numbers and then from some of the principles of Pythagoras. Due to the differences in culture and approach to numbers, different numerologists attach differing attributes to numbers.

Even though there are slight differences here and there in the origin and development of numerology, the centrality of numerology has consistently been to predict the future and outcomes of all kinds of relationships, regulate people’s behavior, have deeper insight into people’s personality, and to built a productive through the aid of numbers unique to individuals, in other words their numerology. For instance, if an individual’s number is 7, it is assumed that the number 7 has its unique vibration and so if the individual in question will be productive with his life, all hi choices must have a corresponding vibration to the number 7. Due to the different schools of thought attributed to the origin and development of numerology, different methods have been used in practicing numerology. However there are three major methods or types of numerology.

The Chaldean numerology: this type or pattern of numerology is popularly considered to be the oldest of all the types of numerology; it originates from the ancient Babylonian kingdom and their practices. The Chaldean numerology is deeply embedded in the culture of the Chaldeans who were occupying Babylon at that time. Although this type of numerology is not commonly used by a lot of people, it is still considered as the most accurate type of numerology. This is due to the fact that it is assumed to have been evidently proven consistent and efficient for thousands of years. The Chaldean system differs from the other two major types of numerology in some important ways. Unlike the other Pythagorean and kabbalah systems for example, the Chaldean system doesn’t really use your original and first name written on your birth certificate. It rather uses the name you are mostly or commonly known by. It can even be your nickname, your marriage name or your real name; so far you are most commonly known by the name. Another interesting difference is that Chaldean numerology doesn’t use single digits alone; it also makes use of double digits. The single digits are used to portray the outer influences in a person’s life while the compound numbers is used to portray someone’s inner aspect. In other words, the single units shows details about you that can be seen by people in the physical world while your double digits represents any unseen or hidden traits or influences in your life that affects your past, present and even your future. The single digits are also know as the physical numbers while the double digits are known as the metaphysical numbers in Chaldean numerology. The single digits have similar meanings but the double digits are much more complex; they have a life of their own.

The name of an individual is the most important factor and has a lot of significance in Chaldean numerology. After the name, the date of birth will now be considered. The date of birth has its occult significance and has a lot of effects on the health, personality and other aspects of an individual’s life. Basically, the Chaldean numerology holds that every letter has a vibration unique to it and numbers assigned to it based on the value attached to its vibration. The numbers in the Chaldean system starts from one and ends in nine, unlike that of the other two systems which is from one to nine. The number nine is considered a holy number and thus is set apart from the other numerical vibrations. However, it is included when it is the sum of all the vibrations.

Kabbalah numerology: this method of numerology does not take account of the numerical value of your birthday, it only uses the person’s full name, words that when said can effect changes on our mood or even our entire life, it provides a remarkable way for people to learn or say, become aware of their true nature, covering into why they behave a certain way, which can even enable them to unlock the mysteries of people around them.

Each person has their peculiar set of numbers related to their names and can be translated to give a personalized numerological chart, the value for the kabbalah numerology is also a factor that differentiates it from other methods, following this factor is the fact that kabbalah also has a much wide range of numbers. An example is that there can be up to 400 different life paths based on 22 different vibrations, which is exhausting and so most numerologists prefer to work with the more accepted life paths numbers. However, because the kabbalah system of numerology uses the full name instead of the birthday number it has been touted to be the less accurate system of numerology when compared to the other two systems by majority of the numerologists.

Pythagorean numerology: Pythagoras numerology is also called modern or western numerology. It is the most widely and commonly used of the three main systems. Its popularity can also be attributed to the fact that it is not ambiguous to interpret letters into digits and master their meanings. The Pythagorean system gives a lot of attention and analysis to the names of individuals. The results of this analysis give a psychological insight to what influences and motivates someone, the natural traits and talents of the person and also predicts certain challenges and/or impacts the person will face or make within his or her environment. With this system people will be able to define the natural talents, tools and abilities inherent in them from birth. And for it to be very effective you have to use your full name given to you at birth.

Ancient Greek Mathematicians

“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, (See Figure 1). 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 (See Figure 2). 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 (See Figure 3).

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 (See Figure 4) (“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 (See Figure 5). 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.

Pythagoras theorem also known as Pythagorean theorem is a quite interesting concept, every Maths student would be familiar with the word, even non-maths students also would have gone through it in their school time. This theorem gives the fundamental aspect in Euclidean Geometry connecting the three sides of a triangle provided the triangle must be right-angled. Geometrically it would be amenable to allude the properties and its various dimensions, the pictorial representation of the theorem, its application to real-life is an accolade. Its geometrical and metric representations are vast and enormous to deal with, it is not possible to derive it in a single post as it requires elaborate information to deal with angles and its functions. Let us see the important facts which allure us to read the post and feel us interesting.

Introduction

Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Ionian Greek philosopher. He started a group of mathematicians who works religiously on numbers and lived like monks. Finally, the Greek Mathematician stated the theorem hence it is called by his name as ‘Pythagoras theorem.’ Though it was introduced many centuries ago its application in the current era is obligatory to deal with pragmatic situations.

Right-angled triangle

A right-angled triangle is a triangle in which one angle is the right angle that is the value of theta which is very common in trigonometry, which is the measurement value of ϴ = 90ᶿ. The side opposite to the right angle is called the hypotenuse side or simply hypot, the other two sides near the right angle are called the opposite side and adjacent side.

Pythagorean triangle and triples

Let us take a right-angled triangle which trifurcates into 3 portions its sides are namely a,b,c. The hypotenuse side is called a, adjacent and opposite sides are called b and c respectively. The hypotenuse side is always the longest side of a triangle which is always consistent in measurement and it occupies a big portion in the triangle.

If the length of all the three sides taken into account of a right-angled triangle are integers then it is called the Pythagorean triangle and the length of the sides a,b,c are collectively known as Pythagorean triples.

Statement of Pythagoras theorem

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the square of the other two sides. (that is adjacent and opposite side)

Let us take three people father, mother, and daughter who are celebrating their only daughter’s birthday with cake cutting event. The square bought by her mother is square. Now the kid cut the cake slantingly. That is four sides are A, B, C, D. She makes cuts on the diagonals that is endpoint A and C. Now the square cake has become two triangles her mother advised her daughter to take the first half(first triangle) and it was shared as follows

The three portions similar to a right-angled triangle a is the hypotenuse side the larger one which the parents gave it to the birthday girl and the remaining portion that is opposite and adjacent are shared between the parents. The same trifurcation can be made for the second half triangle cake portion also.

According to the Pythagoras theorem, the main idea is, if we take measure the sum and squares of b and c side it should be equal to the sum and square of the child’s share that is a.

Mathematically it is stated as follows.

= +

• a – the largest portion, child’s share, hypotenuse side
• b – Father’s share, Opposite side
• c – Mother’s share, Adjacent side

Applications of Pythagoras theorem

Though it is necessary to learn the basic concepts such as theorem statement and its mathematical representation, we would be more curious in understanding the applications of Pythagoras theorem which we decapitate in day to day life situations.

Engineering and Construction fields

Most architects use the technique of Pythagoras theorem to find the value as well as when length or breadth are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields.

Face recognition in security cameras

We are more familiar with face recognition nowadays it reduces the turmoil in investigating the crimes in the security areas. It undergoes the concept of the Pythagorean theorem that is, the distance between the security camera and the place where the person is noted is well projected through the lens using the concept.

Woodworking and interior designing

As the main concept indicates if the cardboards being square can be made into a triangle easily by cutting diagonally then very easily the Pythagoras concept can be applied. Mostly woodworks are made on the strategy which makes the designers easier to proceed.

Navigation

It’s a very amazing fact but people traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.

Surveying

Usually, surveyors use this technique to find the steep mountainous region, knowing the horizontal region it would be easier for them to calculate the rest using the Pythagoras concept. The fixed distance and the varying one can be looked through a telescope by the surveyor which makes the path easier.

Summary

Though the Pythagoras theorem has vast applications very few are mentioned in this blog. Many theorems are stated only with the fundamental concept of the theorem. I think readers after reading this blog would get a clear picture of what Pythagoras theorem is about.

“All men by nature desire to know”Aristotle. That quote sums up how the Greeks looked at thought and learning. They constantly were searching for more knowledge and new ways of discovering it. The Greeks were pioneers in many of the things they did. The Greeks were one of the most influential leaders in math, science, and philosophy, and their ideas are still used today. The common factor between all these things was that in most cases they used deduction, and proof to have a better understanding of themselves and the universe.

The Greeks had many leaders in thought, science, and math. Most of these people weren’t actually just one of these things, they were a combination of many different professions. One of the famous mathematicians of all time was Pythagoras. Pythagoras was a philosopher and a mathematician. He lived from 570 to 490 BCE.y He grew up in Samos, near Turkey, later in life, he lived in southern Italy. Archimedes was an engineer, a physicist, and an inventor. He is most famously an inventor though, having created many inventions that are still used today. He was born in 287 BCE and died in 212 BCE. He was born in Syracuse and moved to Alexandria, but almost nothing is known about his childhood. Thales of Miletus was born in Born in Ionia in 620 BCE, and died in 546 BCE. Thales was a philosopher and a scientist. Ptolemy was an astronomer, geographer, and mathematician. He was born in 127 AD, and died in 145 AD. Almost nothing is known about his life. He wrote a book about astronomy, and he thought the earth was the center of the universe. Euclid, was a teacher, and he ran a school, he also wrote about 12 books, most of them lost. Almost nothing else is known about his life. Plato is one of the most well known philosophers ever. He was born in 427 BCE, and he died in 347 BCE. He taught Aristotle, and was taught by Socrates. Aristotle may be the most well-known philosopher ever, he was also a mathematician, had ideas about politics, a biologist, and a psychologist. He was a student of Plato for 20 years, and rejected some of his theories. He was born in 384 BCE and died in 322 BCE. Aristarchus of Samos was born in 310 BCE and died in 210 BCE. He proposed the idea of the earth being heliocentric, he also said the stars were distant suns that did not move. Ertathosenthes was a philosopher and mathematician, who studied in Athens. He lived from 276-194 BCE. Hipparchus was an astronomer, and a mathematician, who measured the length of a year almost exactly, he was off by only six and a half minutes, he also made the first star catalog, and basic trigonometry. Socrates, although he never wrote anything was one of the most influential philosophers of his time. He was born in 469 BCE and died in 399 BCE. Zeno of Elena is most famous for his work on paradoxes. He studied with a contemporary of Socrates. He lived from 490 BCE to 425 BCE.

The Greeks had many achievements in the field of mathematics. One of these achievements was made by Pythagoras, he discovered how to find the length of sides of right triangles, using The Pythagorean Theorem. The pythagorean theorem states that, a^2, B^2=C^2. The A and B are two shortest sides of the triangle, and taking the square of them both, and adding them, equals the square of the longest side, C. This is still used in architecture, taught in schools, and many more things. The book The Elements, was written by Euclid. He did not create anything new, but rather he combined all the known math at that time, and created one book. This was very useful at the time, because it was very difficult to know all math at that time, but when it was all in one book it became much easier. The book uses only deduction, not any induction. The Greeks were the ones who decided they would use the base number 60. They decided on 60, because it has many factors, because it has many factors, it can easily be divided into segments to make things easier. The Greeks tried to find the square root of two. At that time they did not realize that it was an irrational number, they were determined to find it. Pythagoras was especially interested in this topic. He finally came up with the conclusion, that there was no rational square root of two. ANother problem that dealt with the square root of two, was doubling a cube. At first, they thought they should just double the sides, but that produces a cube eight times the size. To be able to double the size of a cube, you need the square root of two. The Greeks were very exact in their numbers. They were stunned when they realized they couldn’t express all numbers in a fraction.

The Greeks also had many ideas in science. Science is also connected with math. For example Aristacrous was able to roughly measure the size of the sun, moon, and earth.

“We should not give the Greeks too much credit when one of these theories (for example of atoms) anticipates a product of modern science. However, we must give immense credit for the pattern of thought that they introduced, the concept that things could be understood in terms of underlying causes and that these explanations could be tested to see if they were correct. These principles define the scientific method, as we still practice it. Their new method of thought has proven very powerful in making progress in our understanding of nature. Aristotle stated it more clearly than anyone since: ‘There is no science except of the general.’” “The greeks, were not always right, but their process in which they obtained information was ingenious, and still used today. “A powerful aspect of science is that it aims to detach itself from notions with specific use and looks for general principles with broad applications. The more general science becomes the more abstract it is and has more applications.” (Cristian Violatti) Oftentimes the Greeks had ideas, in which they created rules, adn principles. The Greeks had numerous fragments of ideas, and they expanded them into principles. For example, “Egyptians knew, for example, that a triangle whose sides are in a 3:4:5 ratio is a right triangle.” (Cristian Violetta) Pythagoras took that known idea and produced a theorem for all right triangles that is still used today. Archimedes was an inventor, one of his most useful inventions, was his use of the screw. Once a king had a problem, he didn’t know how to take out rainwater from the hull of his ship. He asked Archimedes to help. Archimedes used a screw to lift up the water from the boat. This idea is still used today in some countries, as for types of irrigation.

Introduction

Pythagoras was wrong. The planets in our solar system do not revolve around Earth, nor are they carried in their orbits by crystal spheres. However, his theory of “music of the spheres” holds truths that continue to be uncovered with modern scientific advances. This paper explores the inextricable links between music, science, and faith that are contained within the music of the spheres. Whether or not we believe in the spiritual aspects of the theory is irrelevant; the practical application of these principles affects us every day.

The Music of the Spheres from Antiquity to the Age of Discovery

“The music of the spheres is an ancient concept that the universe is arranged in a logical and orderly manner, consistent with the principles of musical harmony.”[1] The theory involves the planets in our solar system rapidly revolving around Earth, each contained in their own crystal sphere. “Such rapid motion was believed to produce sound, as is commonly experienced when large objects move quickly.”[2] One can envision the scraping of planetary bodies against crystal cages creating all manner of noises. For Greek theorist Pythagoras, who lived from approximately 580 to 500 B.C.E., those sounds formed the structure of music. Mercury, Venus, Mars, Jupiter, Saturn, the sun, and the moon were all visible to ancient scholars who lived before the invention of the telescope. “These seven bodies…were sometimes matched with the seven tones in a musical scale or mode. The eighth sphere was the ‘fixed stars’…The stars thus completed the octave in the planet scale.”[3]

“Although some writers in antiquity seemed to treat the celestial music as actual sounds, most took ‘harmony’ to refer to a logical congruence of elements in the heavens.”[4] The Medieval philosopher Boethius expounded on the idea of music of the spheres in his treatise De Institutione musica, written in the early sixth century. He describes three categories of music, including musica mundana, musica humana, and musica instrumentalis. Musica mundana describes “the numerical relations controlling the movement of the planets, the changing of the seasons, and the combination of elements.”[5] Musica humana “harmonizes and unifies the body and soul and their parts.”[6] Musica instrumentalis, the only category audible to the human ear, includes instrumental and vocal music.

Johannes Kepler, a sixteenth-century German astronomer and mathematician, believed Pythagorean ‘harmony’ to be “in thought, not in sound.”[7] He is known for his three laws of planetary motion and for championing Pythagoras’ theory of ‘celestial music.’[8] His view of ‘heavenly harmony’ posited that “each planet ‘sings’ not a single tone but a range of notes depending on its speed at a particular point in its orbit [around the sun].”[9] The ‘pitch’ of each planet was based on its angular velocity from the sun; as quoted in Rogers, “Venus, having the most nearly circular orbit, remains almost on unison (1:1). Mercury, closest to the sun and with the most eccentric (least circular) orbit, has the widest range (a ratio of 12:5) and the highest ‘voice.’”[10] “Pythagorean ideas about cosmic harmony continued to be elaborated by Neoplatonists from Carolingian times until the end of the Renaissance. These ideas strongly influenced astronomers and astrologers, physicians, architects, humanist scholars and poets.”[11] Isaac Newton, quoted in Rogers, “describes the similarity of the gravitational force to the effect of tension on the strings of a musical instrument: The Sun by his own force acts upon the planets in that harmonic ratio of distances by which the force of tension acts upon strings of different lengths, that is reciprocally in the duplicate ratio of the distances.”[12] He was channeling Pythagoras with this statement, as “Pythagoras showed that a string two feet long would vibrate with a certain tone, and that a string half as long would yield a tone an octave higher.”[13] Thus, the ratio of an octave is 2:1, a perfect fourth is 4:3, a perfect fifth is 3:2, and so on.

The Music of the Spheres in Science Today

With the advent of modern technology, new discoveries are made daily in the fields of science, math, and music. While we know that our planets do not ‘sing,’ we do know that they emit sound far beyond our own hearing abilities. “The new science of helioseismology reveals that the sun itself vibrates with acoustic pressure waves.”[14] Our own planet reverberates with sound as well: “The Schumann Resonances denote a phenomenon that occurs in the Earth’s ionospheric cavity as a result of continual lightning discharges striking the Earth. This ongoing agitation causes the Earth to ring as if it were a giant bell, resulting in a set of quasi-standing waves that measure between approximately 8 Hz and 45 Hz.”[15] Composer David First writes of his fascinating project involving the Schumann Resonances in The Leonardo Music Journal. He was able to network with Davis Sentman of the geophysics department at the University of Alaska, Fairbanks. Sentman records the Schumann Resonances in the area for data to be used in determining global weather patterns; he was able to write a program to send the live data feed to First in Manhattan. Using audio synthesis software, First was able to “transform the real-time data feed into various species of sonic material”[16] with which to compose new music. He writes of the project, named Operation:Kracpot:

My primary method for audification of the Schumann Resonances was to multiply data that was coming in…by a factor of 16, thereby placing the relatively low set of original frequencies into a reasonable musical range- a four-octave transposition. This result was, indeed, a quite beautiful-sounding set of harmonic relationships unlike anything I had previously heard- a transparent, bell-like chordal hum.[17]

First continues his article by explaining that the first bell-like tones that he heard were not represented accurately; he had to account for the “harmonic resonances of a spherical cavity resonator.”[18] In other words, the Earth itself had to be contained in the equation “instead of doing the traditional overtone math.”[19] It was as if someone had left the overtone series out in the rain, and it had warped…There was an emerging pattern revealing all the linear overtone series relationships embedded within the spherical one. They simply were not following a simple line- they were following an ever-widening curve…It begins ‘out of tune’ and rises to a theoretical perfect resolution- albeit one significantly above our range of hearing. But what a beautiful construction to wax poetic about- our imperfect earthly existence reaching perfection somewhere in the heavens.[20] Pythagoras would have been beyond proud.

Overtones and the Music of the Spheres

Overtones are an important part of resonance and must be included in discussion involving music of the spheres. “Every musical note is a composite sound consisting of a fundamental tone, which is usually the pitch we perceive, combined with a number of additional pure tones above it called harmonics or overtones.”[21]

Pythagoras discovered the overtone series, which forms the building blocks of consonant intervals and influences Western music to the present era. The order of the intervals in the overtone series starts with an octave, then a perfect fifth, a perfect fourth, a major third, and the intervals continue to shrink smaller and smaller until all twelve tones of the scale have been heard. This is the reason a perfect fifth and a perfect fourth are considered the most consonant intervals. [22]

The music of the spheres has long inspired belief in a cosmic creator/composer, as evidenced in this statement made by Kepler, quoted in Rogers’ cross-curricular article in the Music Educators Journal: “I feel carried away and possessed by an unutterable rapture over the divine spectacle of the heavenly harmony.”[23] Indeed, Western music has been profoundly influenced and preserved because of the institution of the church. In light of early church beliefs pertaining to the beauty of creation, the linking of consonant, beautiful sounds to the act of worshipping God was inevitable. Delving further into this connection exposes the matter of overtones. “When people hear overtone singing for the first time, the universal reaction is one of amazement. With its otherworldly quality, it is easy to see how the sound of overtone singing is often associated with sacred utterance.”[24]

“The term ‘overtone singing’ refers to techniques that allow a singer to isolate one (or more) of the natural harmonic partials in the overtone series of a sung fundamental pitch, thus making audible two discrete pitches simultaneously.”[25]

It is a vocal technique used by some musicians to strengthen their awareness of overtones, thus enhancing their ability to use these overtones to create a more resonant singing voice. However, Stuart Hinds, a practicing overtone singer, writes that many people use overtone singing for “physical and mental well-being and…as spiritual expression. It should be understood…that for many overtone singers, these holistic and spiritual aspects are the primary reasons for overtone singing.”[26]

The music of the spheres has come full circle, from a means of describing the handiwork of God to a tool with which to praise God for his handiwork.

There are those who use the mystical power of overtones to reach further into human consciousness and even into the human physical form. “Gong sounds affect the human soul and penetrate to deep levels in the subconscious.”[27] Gongs do not have definite pitch; they are aurally identified not by their lowest fundamental, but by their consonant and dissonant overtones. “The overtone series of the various gongs were found to provide satisfactory explanation to the effect of the gongs on patients subjected to music therapy with gongs.”[28] When a gong is played that has strong overtones combining to sound the interval of a diminished sixth, for instance, “people report a pleasant feeling of warmth and security, as if being totally immersed in the sound…In music [the diminished sixth] is considered as a lyrical and consoling interval, and gives a pleasant nostalgic feeling.”[29] The article in the

Journal of New Music Research also recounts an instance of a patient who is cured of chronic pain after listening to repeated playings of Schoenberg’s “Poѐme en Mi” on percussion instruments. The most prominent overtones in the composition consisted of E4, E5, Bflat5, E6, G6, Bflat6, essentially an E diminished chord. “Such personal sensitivity, up to self-identification of a person with an individual note, as if being tuned by it, was reported by Blass in her long experience in work with severely mentally retarded patients.”[30]

Further Applications of the Theory of Music of the Spheres

The theory of music of the spheres has guided music composition and education for thousands of years. Can the intricacies involved therein continue to direct musicians and music educators in 2019?

Fifty years ago, New York’s Union Theological Seminary provided doctoral degrees in church music…today that program is essentially defunct. Just last year…Northwestern University eliminated its degree programs in organ, thus terminating its historic training in the church’s musical art…In many communities, public school music education is a casualty of a pinched budget…Community orchestras are disbanding, and classical music radio broadcasts are being dropped. Perhaps the greatest tragedy is that most younger people today fail to learn to sing- either at the ball game or in church.[31]

Many would agree that the nature of music and of music education is changing. But should we abandon our rich heritage in this age of political correctness just to avoid music’s long history with the church? Donald Paul Hustad, prominent church music historian, does not think so: “I believe that God expects all persons to be ‘ecologists of culture.’”[32] In his article in the Choral Journal, he calls on musicians to preserve our rich history by continuing to contribute to music in church: the driving force behind, and the very reason for, Western music. Many composers (like Beethoven, Schubert, Mendelssohn, and Brahms) contributed masterworks that we still use for worship. That phenomenon demonstrates what theologians call ‘common grace’ – that even unbelieving artists may use God’s creative gifts so well that God takes delight in their works, whether they are offered to God as worship or as competition in creativity.[33]

Pythagoras was wrong. Yet his theory shaped and influenced music, art, architecture, philosophy, science, and religion far beyond his lifetime. Because of the music of the spheres, we experience the mystical, healing, worshipful gift that is Western music today. We must work to keep the spirit of Pythagoras’ theory alive by continuing to create, learn, and collaborate in the fields of science, math, and music.

1. George L. Rogers, “The Music of the Spheres: Cross-Curricular Perspectives on Music and Science.” Music Educators Journal, 2016.
2. Rogers.
3. Rogers.
4. Rogers.
5. Barbara Russano Hanning and J. Peter. Burkholder. Concise History of Western Music. New York: Norton et Company (2014): 22.
6. Hanning, 22.
7. Johannes Kepler. Harmonice Mundi. Translated by E.J. Aiton, A.M. Duncan, and J.V. Field. Phildelphia: American Philosophical Society (1997): 446.
8. Rogers.
9. Rogers.
10. Rogers.
11. James Haar, ‘Music of the spheres.’ Grove Music Online. 2001
12. Rogers.
13. “The Music of the Spheres.” Wilson Quarterly 30, no. 4 (September 2006): 82.
14. Rogers.
15. David First. “The Music of the Sphere…”Leonardo Music Journal 13, no. 1 (December 2003): 32.
16. First, 33.
17. First, 34.
18. First, 34.
19. First, 34.
20. First, 35.
21. Stuart Hinds. “Argument for the Investigation and Use of Overtone Singing.” Journal of Singing 62, no. 1 (September 2005): 33.
22. Richard Cole and Ed Schwartz, eds. “Overtone.” OnMusic Dictionary – Term, June 6, 2016.
23. Rogers.
24. Hinds, 36.
25. Hinds, 33.
26. Hinds, 33.
27. E. Rapoport, S. Shatz, and N. Blass. “Overtone Spectra of Gongs Used in Music Therapy.” JOURNAL OF NEW MUSIC RESEARCH (2008).
28. Rapoport.
29. Rapoport.
30. Rapoport.
31. Donald Paul Hustad. “CREATION, CULTURE, and the ‘MUSIC OF THE SPHERES.’” The Choral Journal 47, no. 9 (2007): 28.
32. Hustad, 29.
33. Hustad, 26.

Pythagoras’s journey began in Samos, Ionia at the time of 570 BCE and he sadly died in Metapontum, Lucanium at the time of 500-490 BCE. Pythagoras got a good quality education because his father (Mnesarchus) was wealthy merchant. He possibly studied in Babylon and Egypt where he may have learnt from the greatest Greek professors. In around 532 BCE, Pythagoras moved to the South of Italy to escape Samos’s cruel orders. Then Pythagoras became one of the most well-known Greek philosophers. Pythagoras changed the perspectives of future philosophers such as Plato and Aristotle.

Pythagoras’s work

Pythagoras achieved many important things in his lifetime. One of his achievements was building a school at Croton. This school was like a secret and undercover brotherhood and monastery. He taught his students about his religious remarks on life. People who joined this society were needed to live properly, love each other, share their beliefs and so on. Pythagoras had many beliefs that made an impact on his achievements. One of his beliefs state that the soul and our thoughts process in the brain instead of the heart.

In the study of mathematics, Pythagoras is known for the Pythagorean Theory. His theory states that in a right-angled triangle, the square of the long side (hypotenuse) is equivalent to the sum of the squares of the other 2 sides. In spite of the Babylonians and the Indians being known to have used this theory, Pythagoras or possibly his former student may have built the first proof. He discovered mathematical proportions, square numbers and square roots. The finding of the golden ratio is sometimes also credited to Pythagoras or maybe his student named by Theano. He was also credited to be one of the first to think that the Earth was round, that planets have an axis that they rotate on and that planets circulate around one main point. Pythagoras also had a love for music. According to pastimes, he found out that musical notes could possibly be made into mathematical equations. Therefore, Pythagoras has achieved many in his journey through the world of mathematics, religion, philosophy, science, music and possibly even more.

Citations

1. Mastin L. ‘Pythagoras > By Individual Philosopher > Philosophy.’ Jan. 2009, Accessed 27 Feb. 2020. https://www.philosophybasics.com/philosophers_pythagoras.html
2. Math Is Fun, and Rod Pierce. ‘Definition of Pythagoras Theorem.’ 27 July 2018, Accessed 27 Feb. 2020. https://www.mathsisfun.com/definitions/pythagoras-theorem.html

When the empire of the Greek began to spread all over the world especially into Asia, the Greeks were so clever and smart that they could adopt and adapt useful factors or elements from the communities they invaded. In fact they adapted many elements of mathematics from both the Babylonians and the Egyptians. However , the Greek began at once to develop and to make important contributions in the field of mathematics . One of these contributions is one of the most outstanding revolutions on mathematics during the Hellenistic Period .

The Hellenistic period had a period in ancient history in which Greek culture was rich in many aspects of civilization at that time. It started after the death of Alexander the Great in 323 BC, and lasted about 200 years in Greece and about 300 years in the Middle East

What are these contributions ?

Philosophy and Mathematics

The most famous figure of the 6th Century BCE mathematicians was definitely Pythagoras of Samos , a Greek island in the Aegean Sea, close to the coast of western Turkey. Indeed, Pythagoras is believed to be the first to coin both the words “philosophy” (“love of wisdom“) and “mathematics” . He was perhaps the first to know that geometric elements corresponded with numbers so he could construct or build a complete system of mathematics Pythagoras’s Theorem (or the Pythagorean Theorem) is one of the best known of all mathematical theorems.

Thales’ Intercept Theorem

Geometry was the most relevant aspect that controlled the Greek Mathematics . One of the cleverest mathematicians was Thales of Miletus . He was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He is usually considered to be the first to lay down guidelines for the abstract development of geometry.

Thales founded what has become known as Thales’ Theorem. This theory says if a triangle is drawn within a circle with the long side as a diameter of the circle, then the opposite angle will always be a right angle. Besides , he is known as Thales’ Theorem or the Intercept Theorem, about the ratios of the line segments that are created if two intersecting lines are intercepted by a pair of parallels (and, by extension, the ratios of the sides of similar triangles).

Attic or Herodianic numerals

The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed by about 450 BCE, and in regular use possibly as early as the 7th Century BCE. It was a base 10 system similar to the earlier Egyptian one (and even more similar to the later Roman system), with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the desired number. Addition was done by totaling separately the symbols (1s, 10s, 100s, etc) in the numbers to be added, and multiplication was a laborious process based on successive doublings (division was based on the inverse of this process).

Three geometrical problems

These three geometrical problems were profoundly influential and effective on future geometry and paved the way to many useful and outstanding discoveries . Despite the fact that their actual solutions had to wait until the nineteenth Century . Three geometrical problems in particular could be solved by purely geometric means using only a straight edge and a compass, date back to the early days of Greek geometry: “the squaring (or quadrature) of the circle”, “the doubling (or duplicating) of the cube” and “the trisection of an angle”. Hippocrates of Chios was one such Greek mathematician who applied himself to these problems during the 5th Century BCE .His influential book “The Elements”, dating to around 440 BCE, was the first compilation of the elements of geometry, and his work was an important source for Euclid‘s later work.

Zeno’s Paradox of Achilles and the Tortoise

The most famous of his paradoxes is that of Achilles and the Tortoise, which describes a theoretical race between Achilles and a tortoise. Achilles gives the much slower tortoise a head start, but by the time Achilles reaches the tortoise’s starting point, the tortoise has already moved ahead. By the time Achilles reaches that point, the tortoise has moved on again, etc, etc, so that in principle the swift Achilles can never catch up with the slow tortoise. These paradoxes are based on the infinite divisibility of space and time, and rest on the idea that a half plus a quarter plus an eighth plus a sixteenth, etc, etc, to infinity will never quite equal a whole.

As just seen through the previous examples , the most important contributions of the Greeks , especially the figures Pythagoras, Plato and Aristotle , were all influential . They used the idea of the proof and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Plato who is best known for his description of the five Platonic solids established his famous Academy in Athens in 387 BCE . Aristotle’s work on logic was regarded as definitive for over two thousand years.