CAMILLE ZEITOUNI'S INTEGRATED PROJECT |
The roots of mathematics start with the concept of quantity, magnitude and geometry. To start off, the concept of quantity comes from the hunter-gatherer society. As proof, there are still hunter-gatherer societies whose language only includes the numbers one, two and many to quantify. Prehistoric artifacts found in Africa, show proof of humans trying to quantity time. For example, the Lebombo bone described as the oldest mathematic artifact, shows 29 tallies, which could be interpreted as a lunar cycle. According to the Universal Book of Mathematics, this may make “African women the first mathematicians because keeping track of their menstrual cycles requires a lunar calendar.” Pre-dynastic Egyptians and Sumerians used geometric design on various artifacts. However, mathematics did not officially first appear until civilizations settled and developed agriculture.
The oldest undisputed use of mathematics is in Babylonian and dynastic Egypt. Sumerian and Babylonian mathematics is based on a base 60 number system (because of its large number of divisors). Meaning they had numbers from 1 to 59 (1 and 60 are represented by the same representation) and would calculate other number using those numbers. They also had a true place value system, where digits in the left column were larger in value than the ones towards right. This is also a manifestation of magnitude. For example, would equal to (60x60=3600)+(60x1)+(1)= 3661. (See Visual Aids) This is cool but you may ask: “how does this have anything to do with me?” Let me explain. It has been conjectured that 60 seconds in a minute, 60 minutes in an hour and 360 degrees in a circle and the 12 months in a year, 12 inches, 12 pence, 2x12=24 hours in a day (60/5=12) are all mathematic concepts left to us today by the Babylonians.
As for the Egyptians, they used a base 10 system but did not have a place value system meaning you would need 54 characters to represent one million minus 1 (they had a representation for one million). An important artifact left to us by the Egyptians is the Rhind Papyrus. It’s a set of instructions in arithmetic and geometry. It can tell us how to multiply, divide, use unit fractions, solve first order linear equations as well as arithmetic and geometric series and gives us knowledge on composite numbers and prime numbers. The Egyptians could also solve second-order algebraic equations. Another interesting fact was that Egyptians knew how to approximately calculate the area of a circle. They did this by using shapes whose areas they already knew. They found that they could calculate the area by multiplying its diameter by 8/9 and then squaring it. This effective is π within less that one percent. When we think of Egypt, we think of pyramids. This is also an indication of mathematics because we can observe the golden ratio (1:1.618). This could have also just occurred for aesthetic reasons but there is evidence that they knew how to find the volume of a pyramid and were aware that 3,4 and 5 triangles (will be explained later when we look at Pythagoras) gave a right angle. While all this sets the base of mathematics, let’s go a little closer in time and examine what happened next.
Ancient Greece is said to be the culture, which provided foundation of modern Western culture. The many philosophers and the initial development of democracy characterize this time period. There were many advances in mathematics during this time period. Greek mathematics refers to the mathematics written in the Greek language. To understand the mathematics during the time period we must look at the different mathematicians.
Thales of Miletus is thought to have started Greek mathematics. In the formal sense, Thales did not invent anything or formulate proofs. Instead, he noticed patterns and he put forth conjectures that he “proved” through repeated demonstrations. If there were no contradictions, he would justify them as true. For example, he states in his proposition I.26: “If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle”. He used this to calculate the height of the Great Pyramid, which was still not known for 2000 years. He measured the height using the shadow the pyramid had cast. This is the first introduction of the concepts of ratio and proportionality.
Next, we have the famous and controversial Pythagoras. I will begin with the controversial aspect. First and foremost, he left no mathematical writings himself thus we don’t know if the theorems were really proved by him. Second, Pythagoras was the head of a school (“cult”) that had many followers called the Pythagoreans. The Pythagorean sect was profoundly mythical; it largely dominated by mathematics but also had many religious philosophies such as strict vegetarianism, communal living, secrets rituals and some odd rules such as never urinating towards the sun, never marrying a woman who wore gold jewelry, never eating or touching a black fava bean… The cult’s principle philosophy was “All is number” or “God is number” and the Pythagoreans practiced number worship. This means they gave numbers gender (women were odd numbers and men even numbers) and each number had its own character. For instance, one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets; etc. The sect ended when the two sub-divisions of the group (the Mathematikoi and the Akousmatikoi) got in a huge fight, along with the resentment of secrecy and exclusiveness that the school was burnt down and destroyed. Secondly, I will elaborate on why Pythagoras is so famous. Pythagoras is said to have discovered the Pythagorean theorem (a2+b2=c2) and thus the Pythagorean triplets. For example, 32+42=52. This also led to the appearance of irrational numbers. For example, try using the Pythagorean theorem with sides of 1. You get √2 which one of students Hippasus tried to express as a fraction but could not. This was a big deal because Pythagoras’ cult was built on the elegance of mathematics meaning the existence of a number could be expressed as a ratio of two of God’s creations and shattered their entire belief system. Thus Pythagoras kept this secret and thus Hippasus never got to share his discovery at the time with the world. Pythagoras or his followers are credited with realizing that the sum of the angles of a triangle are 180o, established the foundation of number theory and found the first pair of amicable numbers. Pythagoras and/or the Pythagoreans were also musicians. They discovered that intervals between harmonious numbers are always a whole number ratio. All this is cool but how does knowing that “for any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the other two sides” help me with anything? It might not but it can be used in architecture and construction, navigation, earthquake analysis and the one I found most interesting crime scene investigation. Forensic investigators determine the path of a bullet took before impact using the Pythagorean theorem and thus pin point its origin, which can help in numerous ways. Blood spatter is also analyzed with the Pythagorean theorem and use this to position both the victim and the assailant. It may not have such a direct impact on our lives but it is still a huge discovery.
Let’s move on to Euclid, “The Father of Geometry”. Euclid wrote the most influential non-religious book named The Elements. It was separated into thirteen smaller books. In the books one to four and six, he talks about plane geometry. In books eleven to thirteen, he talks about solid geometry. In books five and ten, the subject is magnitudes and ratios. In books seven to nine, he talks about whole number. Euclid did not come up with everything in his books but he presented them in a single, simple, logical progression and this facilitated its use and its ability to be reference. He used an approach called the “synthetic approach” to show his theorems. This method progresses from the known to the unknown in logical steps. Euclid came up with ten axioms, which we believe to be true but cannot prove. He separated them in two groups. The first were common notions, an example would be “Things that are equal to the same thing are also equal to each other” and the second group were specific to geometry and an example of that would be “All right angles are congruent”. Euclid’s influence was so large owing to the fact that his work was more than a book about mathematics; it was the way he organized such book and demanded proof for every theorem. This shaped the ideas of western philosophers right until even today. An example of this influence was Newton who presented his work using Euclid’s structure and format and Abraham Lincoln who based the US Declaration of Independence on Euclid’s axiomatic system. This shows that mathematicians can have more influence than just solely in a mathematic field. The Elements was used as a textbook for about two thousand years.
After being taught by the followers of Euclid, Apollonius went on to write the book of conics where he introduced terms such as parabola, ellipse and hyperbola. There were eight books. He introduced the basics then went on to form more complex theorems not included in The Elements.
Now let’s move on to another great mathematician: Archemedes. Archemedes had a passion for pure mathematics. He came up with a new method of calculating areas of regular shapes by inscribing them in bigger and smaller shapes that he already understood. He used a method of exhaustion also known as “Archimedes Method” to come up with a value for π. He estimated between 31/7 and 310/11. Another example of his meticulous work is his calculation of √3 being between 265/153 and 1351/780. Archimedes also discovered the relationship between the volume of a sphere and cylinder and the surface of a sphere and cylinder of equal height and diameter was two-thirds. An interesting thing about Archimedes was that his last words are thought to have been “Do not disturb my circles!” before a Roman soldier killed him after being enraged because Archimedes would not go meet the Roman general until he was done his math problem.
Lastly, let’s discuss “The Father of Algebra”: Diaphanous. Diaphanous wrote a series of books called the Arithmetica, which is a collection of algebraic equations later used to develop number theory. He was also the first mathematician to recognize fractions as numbers. Diaphanous’ general approach to algebraic problems was to determine if a problem has a set of answers or infinity amount of solutions or none at all. Algebra is something used in everyday life. Sometimes we don’t even realize it. For example, you have 12 items to put into 3 bags. How many items do you put per bag? Four and how do you know that? Algebra. Greek mathematics was and still is so influential because it introduced the idea of proof. After all this brilliance, let’s move on to the next time period: The Middle Ages.
To understand what was happening in mathematics during The Middle Ages, we must look at it in a geographical way. Starting off in Europe. To summarize what was happening in Europe at the time: not much. The Dark Ages in Europe started after the fall of the Roman Empire. It is characterized by the strong presence of religion and monarchy and thus science, mathematics and almost all intellectual endeavors stopped. One of the only interesting medieval mathematicians was Fibonacci (actual name was Leonardo of Pisa). He is known for the Fibonacci sequence of numbers but more importantly he brought the use of Hindu-Arabic numeral system to Europe. Even though Europe was in an intellectual downfall. Other parts of the world were still doing very well.
To start off, Indian mathematics was developing independently of Europe and China. Some great mathematical developments come from Brahmagupta. He established basic mathematical rules dealing with zero. For example, 1+0=0, 1-0=1, 1x0=0. He also came up with rules for equations with negative numbers and showed that quadratic equations could have a negative solution and a positive solution. Bhaskara II came later and added to Brahmmagupta’s rules dealing with zero and established and explained that 1/0=∞. The notion of zero is important and can be observed anywhere. If you have 3 lollipops and I take away 3 then you have none. If you take that nothing and multiply it by 3, you still have nothing. The concept of zero is something that you use without even realizing. It is also seen that Indian astronomers used trigonometry to calculate distances from the Moon, Earth and Sun. They calculated that “when the Moon is half full and opposes the sun, then the sun, moon and Earth make a 90o angle triangle. Using this, they found that the Sun is four hundred times further away than the Moon. Another important discovery at the time was by Aryabhata who came to the awareness that π was an irrational number, which was not proven until much later in Europe.
Next, let’s take a trip to China. The Chinese represented their numbers from one to nine. (See Visual Aids) They then placed them in columns that represent units, tens, and hundreds… The Chinese had no symbol for zero, which made it hard to write large numbers. There was a great fascination for number patterns. For example, The Lo Shu Square (See Visual Aids) was seen as having spiritual and religious meaning. The main component of Chinese mathematics was problem solving that could be applied in the civilization such as trade, taxation, engineering and payment of wages. They used a textbook called “Jiuzhang Suanshu” or “Nine Chapters” written by Liu Hui, which told them how to solve equation by using known information to find unknown information. This later appeared in the West when Carl Friedrich Gauss “discovered it”.
Lastly, let’s go see what was happening in the Islamic World during this time. The Islamic Empire, which is geographically between Europe and Asian, fused together the mathematical ideas from ancient Greece and India to form their own mathematics. For example, because they were not allowed to depict humans, they used geometric patterns to decorate buildings. This made mathematics a sort of art form. They used concepts such as symmetry to form this art. Much of the Greek mathematics was translated into Arabic and thus many records are still present because of this translation. An interesting fact is that algebra is Arabic and means reunion of broken parts. An important Persian mathematician named Muhammad Al-Karaji worked to further our knowledge on algebra and introduced algebraic calculus. He was also a strong fan of the Hindu numeral system and pushed for it to be adopted in the Islam world, which it did and later went on to Europe as well. Another great mathematician was Nasir Al-Din Al-Tusi who contributed the law of sines: a⁄(sin A) = b⁄(sin B) = c⁄(sin C). Islamic mathematics stopped because of the influence of the Turkish Ottoman Empire and more developments were moved back to Europe who started to get itself out of its Dark Age. Let’s go check that out.
The Renaissance was a time period of great time of learning based on classical sources. Much of the ancient Greek mathematics was taken and translated and relearnt. An interesting aspect of mathematics was its relevancy to astronomy. It is because of mathematics that we know so much about astronomy. Tycho Brahe was one of the last major naked eye astronomers. He came up with the Tychonic system using his knowledge of geometry and his observations of planets, which is a geocentric model thus he was false. In his later life, Brahe assisted Johannes Kepler. Johannes Kepler came up with the three laws of planetary motion. A more mathematical development was his New Year’s gift to a friend. It was a short pamphlet called “A New Year’s Gift of Hexagonal Snow” and he described hexagonal symmetry of snowflakes and a hypothesis for symmetry (Kepler conjecture), which explains the most efficient arrangement for packing spheres. He also pioneered the mathematic application of infinitesimals (1/∞).
Another brilliant discovery was by John Napier and it was logarithms. His work had fifty-seven pages of explanatory matter and ninety pages of tables of numbers that related to natural logarithms. It was called: Mirifici Logarithmorum Canonis Descriptio (Scottish). Logarithms can be found in nature in seashells with perfect logarithmic spirals. (See Visual Aids) There are also many logarithmic scales such as sound in decibels and the Richter scale for earthquakes.
Calculus came also at this time and was independently invented by Gottfried Leibniz and Newton at the same time. Calculus is a branch of mathematics that finds properties of derivatives and integrals functions. It does this by basing its summation on infinitesimal differences.
Another mathematical advance was Pierre de Fermat and Blaise Pascal who started with the mathematical treatment of dice, which probability was later applied to by Bernoulli and Abraham de Moivre. Probability is a math very commonly used in our everyday life. Sports and board games use probability to determine possible outcomes of games. We use probability along with common sense to make decisions. For example, when you leave your house in the morning, you think about the probability that it will rain based on the weather forecast and decide whether to bring your umbrella or not. That is an application of mathematics you don’t even realize you are doing.
Another important mathematician of the time was Euler. He is the only mathematician to have two numbers named after him! In calculus: Euler’s Number e (2.71828…) and the Euler-Mascheroni Constant γ (gamma) also known as Euler’s constant. It is approximately 0.57721 and it is not known whether it is irrational or not. He also came up with the concept of a function (f(x)). He disproved the first theorem of graph theory.
In brief, after a period of little to no activity in Europe, there was suddenly a huge explosion of knowledge and discovery and mathematics developed immensely. Since then there have been more discoveries and developing in mathematics at a constant rate and more is known as each day passes.
Let’s summarize, we’ve started from the very being in those hunter-gatherer societies who quantify using one, two or many. We have discovered the roots of mathematics coming from Babylon and dynastic Egypt. Then we looked at some of the great ancient Greek mathematicians such as Thales, Pythagoras, Euclid, Apollonius, Archemedes and Diaphanous. After, we left Europe and discovered what was happening in India, China and the Islamic World. We ended in Europe in the Renaissance and discovered the many brilliant developments. However, it is not over. Mathematics is a field that is constantly developing. Each day more theorems are being proven or disproved. The more recent mathematics quests are for the Millennium Prize Problems. In the year 2000, the Clay Institute stated seven problems that remained unsolved and has promised a reward of 1,000,000$ for whoever can solve one. Grigori Perelman solved the Poincaré Conjecture in 2003. He declined the reward deeming it unfair because of the work of Richard Hamilton had done that was the fundamental base of his proof. There are still six Millennium Prize Problems to solve. The history of mathematics is not over.
The oldest undisputed use of mathematics is in Babylonian and dynastic Egypt. Sumerian and Babylonian mathematics is based on a base 60 number system (because of its large number of divisors). Meaning they had numbers from 1 to 59 (1 and 60 are represented by the same representation) and would calculate other number using those numbers. They also had a true place value system, where digits in the left column were larger in value than the ones towards right. This is also a manifestation of magnitude. For example, would equal to (60x60=3600)+(60x1)+(1)= 3661. (See Visual Aids) This is cool but you may ask: “how does this have anything to do with me?” Let me explain. It has been conjectured that 60 seconds in a minute, 60 minutes in an hour and 360 degrees in a circle and the 12 months in a year, 12 inches, 12 pence, 2x12=24 hours in a day (60/5=12) are all mathematic concepts left to us today by the Babylonians.
As for the Egyptians, they used a base 10 system but did not have a place value system meaning you would need 54 characters to represent one million minus 1 (they had a representation for one million). An important artifact left to us by the Egyptians is the Rhind Papyrus. It’s a set of instructions in arithmetic and geometry. It can tell us how to multiply, divide, use unit fractions, solve first order linear equations as well as arithmetic and geometric series and gives us knowledge on composite numbers and prime numbers. The Egyptians could also solve second-order algebraic equations. Another interesting fact was that Egyptians knew how to approximately calculate the area of a circle. They did this by using shapes whose areas they already knew. They found that they could calculate the area by multiplying its diameter by 8/9 and then squaring it. This effective is π within less that one percent. When we think of Egypt, we think of pyramids. This is also an indication of mathematics because we can observe the golden ratio (1:1.618). This could have also just occurred for aesthetic reasons but there is evidence that they knew how to find the volume of a pyramid and were aware that 3,4 and 5 triangles (will be explained later when we look at Pythagoras) gave a right angle. While all this sets the base of mathematics, let’s go a little closer in time and examine what happened next.
Ancient Greece is said to be the culture, which provided foundation of modern Western culture. The many philosophers and the initial development of democracy characterize this time period. There were many advances in mathematics during this time period. Greek mathematics refers to the mathematics written in the Greek language. To understand the mathematics during the time period we must look at the different mathematicians.
Thales of Miletus is thought to have started Greek mathematics. In the formal sense, Thales did not invent anything or formulate proofs. Instead, he noticed patterns and he put forth conjectures that he “proved” through repeated demonstrations. If there were no contradictions, he would justify them as true. For example, he states in his proposition I.26: “If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle”. He used this to calculate the height of the Great Pyramid, which was still not known for 2000 years. He measured the height using the shadow the pyramid had cast. This is the first introduction of the concepts of ratio and proportionality.
Next, we have the famous and controversial Pythagoras. I will begin with the controversial aspect. First and foremost, he left no mathematical writings himself thus we don’t know if the theorems were really proved by him. Second, Pythagoras was the head of a school (“cult”) that had many followers called the Pythagoreans. The Pythagorean sect was profoundly mythical; it largely dominated by mathematics but also had many religious philosophies such as strict vegetarianism, communal living, secrets rituals and some odd rules such as never urinating towards the sun, never marrying a woman who wore gold jewelry, never eating or touching a black fava bean… The cult’s principle philosophy was “All is number” or “God is number” and the Pythagoreans practiced number worship. This means they gave numbers gender (women were odd numbers and men even numbers) and each number had its own character. For instance, one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets; etc. The sect ended when the two sub-divisions of the group (the Mathematikoi and the Akousmatikoi) got in a huge fight, along with the resentment of secrecy and exclusiveness that the school was burnt down and destroyed. Secondly, I will elaborate on why Pythagoras is so famous. Pythagoras is said to have discovered the Pythagorean theorem (a2+b2=c2) and thus the Pythagorean triplets. For example, 32+42=52. This also led to the appearance of irrational numbers. For example, try using the Pythagorean theorem with sides of 1. You get √2 which one of students Hippasus tried to express as a fraction but could not. This was a big deal because Pythagoras’ cult was built on the elegance of mathematics meaning the existence of a number could be expressed as a ratio of two of God’s creations and shattered their entire belief system. Thus Pythagoras kept this secret and thus Hippasus never got to share his discovery at the time with the world. Pythagoras or his followers are credited with realizing that the sum of the angles of a triangle are 180o, established the foundation of number theory and found the first pair of amicable numbers. Pythagoras and/or the Pythagoreans were also musicians. They discovered that intervals between harmonious numbers are always a whole number ratio. All this is cool but how does knowing that “for any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the other two sides” help me with anything? It might not but it can be used in architecture and construction, navigation, earthquake analysis and the one I found most interesting crime scene investigation. Forensic investigators determine the path of a bullet took before impact using the Pythagorean theorem and thus pin point its origin, which can help in numerous ways. Blood spatter is also analyzed with the Pythagorean theorem and use this to position both the victim and the assailant. It may not have such a direct impact on our lives but it is still a huge discovery.
Let’s move on to Euclid, “The Father of Geometry”. Euclid wrote the most influential non-religious book named The Elements. It was separated into thirteen smaller books. In the books one to four and six, he talks about plane geometry. In books eleven to thirteen, he talks about solid geometry. In books five and ten, the subject is magnitudes and ratios. In books seven to nine, he talks about whole number. Euclid did not come up with everything in his books but he presented them in a single, simple, logical progression and this facilitated its use and its ability to be reference. He used an approach called the “synthetic approach” to show his theorems. This method progresses from the known to the unknown in logical steps. Euclid came up with ten axioms, which we believe to be true but cannot prove. He separated them in two groups. The first were common notions, an example would be “Things that are equal to the same thing are also equal to each other” and the second group were specific to geometry and an example of that would be “All right angles are congruent”. Euclid’s influence was so large owing to the fact that his work was more than a book about mathematics; it was the way he organized such book and demanded proof for every theorem. This shaped the ideas of western philosophers right until even today. An example of this influence was Newton who presented his work using Euclid’s structure and format and Abraham Lincoln who based the US Declaration of Independence on Euclid’s axiomatic system. This shows that mathematicians can have more influence than just solely in a mathematic field. The Elements was used as a textbook for about two thousand years.
After being taught by the followers of Euclid, Apollonius went on to write the book of conics where he introduced terms such as parabola, ellipse and hyperbola. There were eight books. He introduced the basics then went on to form more complex theorems not included in The Elements.
Now let’s move on to another great mathematician: Archemedes. Archemedes had a passion for pure mathematics. He came up with a new method of calculating areas of regular shapes by inscribing them in bigger and smaller shapes that he already understood. He used a method of exhaustion also known as “Archimedes Method” to come up with a value for π. He estimated between 31/7 and 310/11. Another example of his meticulous work is his calculation of √3 being between 265/153 and 1351/780. Archimedes also discovered the relationship between the volume of a sphere and cylinder and the surface of a sphere and cylinder of equal height and diameter was two-thirds. An interesting thing about Archimedes was that his last words are thought to have been “Do not disturb my circles!” before a Roman soldier killed him after being enraged because Archimedes would not go meet the Roman general until he was done his math problem.
Lastly, let’s discuss “The Father of Algebra”: Diaphanous. Diaphanous wrote a series of books called the Arithmetica, which is a collection of algebraic equations later used to develop number theory. He was also the first mathematician to recognize fractions as numbers. Diaphanous’ general approach to algebraic problems was to determine if a problem has a set of answers or infinity amount of solutions or none at all. Algebra is something used in everyday life. Sometimes we don’t even realize it. For example, you have 12 items to put into 3 bags. How many items do you put per bag? Four and how do you know that? Algebra. Greek mathematics was and still is so influential because it introduced the idea of proof. After all this brilliance, let’s move on to the next time period: The Middle Ages.
To understand what was happening in mathematics during The Middle Ages, we must look at it in a geographical way. Starting off in Europe. To summarize what was happening in Europe at the time: not much. The Dark Ages in Europe started after the fall of the Roman Empire. It is characterized by the strong presence of religion and monarchy and thus science, mathematics and almost all intellectual endeavors stopped. One of the only interesting medieval mathematicians was Fibonacci (actual name was Leonardo of Pisa). He is known for the Fibonacci sequence of numbers but more importantly he brought the use of Hindu-Arabic numeral system to Europe. Even though Europe was in an intellectual downfall. Other parts of the world were still doing very well.
To start off, Indian mathematics was developing independently of Europe and China. Some great mathematical developments come from Brahmagupta. He established basic mathematical rules dealing with zero. For example, 1+0=0, 1-0=1, 1x0=0. He also came up with rules for equations with negative numbers and showed that quadratic equations could have a negative solution and a positive solution. Bhaskara II came later and added to Brahmmagupta’s rules dealing with zero and established and explained that 1/0=∞. The notion of zero is important and can be observed anywhere. If you have 3 lollipops and I take away 3 then you have none. If you take that nothing and multiply it by 3, you still have nothing. The concept of zero is something that you use without even realizing. It is also seen that Indian astronomers used trigonometry to calculate distances from the Moon, Earth and Sun. They calculated that “when the Moon is half full and opposes the sun, then the sun, moon and Earth make a 90o angle triangle. Using this, they found that the Sun is four hundred times further away than the Moon. Another important discovery at the time was by Aryabhata who came to the awareness that π was an irrational number, which was not proven until much later in Europe.
Next, let’s take a trip to China. The Chinese represented their numbers from one to nine. (See Visual Aids) They then placed them in columns that represent units, tens, and hundreds… The Chinese had no symbol for zero, which made it hard to write large numbers. There was a great fascination for number patterns. For example, The Lo Shu Square (See Visual Aids) was seen as having spiritual and religious meaning. The main component of Chinese mathematics was problem solving that could be applied in the civilization such as trade, taxation, engineering and payment of wages. They used a textbook called “Jiuzhang Suanshu” or “Nine Chapters” written by Liu Hui, which told them how to solve equation by using known information to find unknown information. This later appeared in the West when Carl Friedrich Gauss “discovered it”.
Lastly, let’s go see what was happening in the Islamic World during this time. The Islamic Empire, which is geographically between Europe and Asian, fused together the mathematical ideas from ancient Greece and India to form their own mathematics. For example, because they were not allowed to depict humans, they used geometric patterns to decorate buildings. This made mathematics a sort of art form. They used concepts such as symmetry to form this art. Much of the Greek mathematics was translated into Arabic and thus many records are still present because of this translation. An interesting fact is that algebra is Arabic and means reunion of broken parts. An important Persian mathematician named Muhammad Al-Karaji worked to further our knowledge on algebra and introduced algebraic calculus. He was also a strong fan of the Hindu numeral system and pushed for it to be adopted in the Islam world, which it did and later went on to Europe as well. Another great mathematician was Nasir Al-Din Al-Tusi who contributed the law of sines: a⁄(sin A) = b⁄(sin B) = c⁄(sin C). Islamic mathematics stopped because of the influence of the Turkish Ottoman Empire and more developments were moved back to Europe who started to get itself out of its Dark Age. Let’s go check that out.
The Renaissance was a time period of great time of learning based on classical sources. Much of the ancient Greek mathematics was taken and translated and relearnt. An interesting aspect of mathematics was its relevancy to astronomy. It is because of mathematics that we know so much about astronomy. Tycho Brahe was one of the last major naked eye astronomers. He came up with the Tychonic system using his knowledge of geometry and his observations of planets, which is a geocentric model thus he was false. In his later life, Brahe assisted Johannes Kepler. Johannes Kepler came up with the three laws of planetary motion. A more mathematical development was his New Year’s gift to a friend. It was a short pamphlet called “A New Year’s Gift of Hexagonal Snow” and he described hexagonal symmetry of snowflakes and a hypothesis for symmetry (Kepler conjecture), which explains the most efficient arrangement for packing spheres. He also pioneered the mathematic application of infinitesimals (1/∞).
Another brilliant discovery was by John Napier and it was logarithms. His work had fifty-seven pages of explanatory matter and ninety pages of tables of numbers that related to natural logarithms. It was called: Mirifici Logarithmorum Canonis Descriptio (Scottish). Logarithms can be found in nature in seashells with perfect logarithmic spirals. (See Visual Aids) There are also many logarithmic scales such as sound in decibels and the Richter scale for earthquakes.
Calculus came also at this time and was independently invented by Gottfried Leibniz and Newton at the same time. Calculus is a branch of mathematics that finds properties of derivatives and integrals functions. It does this by basing its summation on infinitesimal differences.
Another mathematical advance was Pierre de Fermat and Blaise Pascal who started with the mathematical treatment of dice, which probability was later applied to by Bernoulli and Abraham de Moivre. Probability is a math very commonly used in our everyday life. Sports and board games use probability to determine possible outcomes of games. We use probability along with common sense to make decisions. For example, when you leave your house in the morning, you think about the probability that it will rain based on the weather forecast and decide whether to bring your umbrella or not. That is an application of mathematics you don’t even realize you are doing.
Another important mathematician of the time was Euler. He is the only mathematician to have two numbers named after him! In calculus: Euler’s Number e (2.71828…) and the Euler-Mascheroni Constant γ (gamma) also known as Euler’s constant. It is approximately 0.57721 and it is not known whether it is irrational or not. He also came up with the concept of a function (f(x)). He disproved the first theorem of graph theory.
In brief, after a period of little to no activity in Europe, there was suddenly a huge explosion of knowledge and discovery and mathematics developed immensely. Since then there have been more discoveries and developing in mathematics at a constant rate and more is known as each day passes.
Let’s summarize, we’ve started from the very being in those hunter-gatherer societies who quantify using one, two or many. We have discovered the roots of mathematics coming from Babylon and dynastic Egypt. Then we looked at some of the great ancient Greek mathematicians such as Thales, Pythagoras, Euclid, Apollonius, Archemedes and Diaphanous. After, we left Europe and discovered what was happening in India, China and the Islamic World. We ended in Europe in the Renaissance and discovered the many brilliant developments. However, it is not over. Mathematics is a field that is constantly developing. Each day more theorems are being proven or disproved. The more recent mathematics quests are for the Millennium Prize Problems. In the year 2000, the Clay Institute stated seven problems that remained unsolved and has promised a reward of 1,000,000$ for whoever can solve one. Grigori Perelman solved the Poincaré Conjecture in 2003. He declined the reward deeming it unfair because of the work of Richard Hamilton had done that was the fundamental base of his proof. There are still six Millennium Prize Problems to solve. The history of mathematics is not over.