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Geometry History Project

Contributions of Mathematicians to the study of geometry
by

Morgan Kuryla

on 26 October 2012

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Transcript of Geometry History Project

History Timeline: Mathematicians and Concepts Thales: 624-547 BC Pythagoras: 569-475 BC BC Hippocrates of Chios: 470-410 BC Eudoxus: 408-355 BC Euclid: 326-265 BC Archimedes: 287-212 BC Hipparchus: 190-120 BC Heron: 10-75 AD Hypatia: 370-415 AD Ptolemy: 85-165 AD Proclus: 411-485 AD Aryabhata I: 476-550 AD 1-599 AD 600-1199 AD 1200-1699 AD 1700-Present Bhaskara I: 600-680 AD Al Battani: 850-929 AD Abu'l Wafa: 940-998 AD Khayyam, Omar: 1038-1131 AD Al Biruni: 973-1048 AD Al Tusi, Nasir: 1201-1274 AD Descartes: 1596-1650 AD Saccheri: 1667-1733 AD Lambert: 1728-1777 AD Legendre: 1753-1833 AD Playfair: 1748-1819 AD Hilbert: 1862-1943 AD Klein, Felix: 1849-1925 AD Riemann: 1826-1866 AD Wantzel: 1814-1848 AD Bolyai, Janos: 1802-1860 AD Lobachevsky: 1792-1865 AD Gauss: 1777-1855 AD Beltrami: 1835-1900 AD Measurement Early Algebra Number Systems Probability and Statistics Discrete Math
and Proof Calculus Traditional Geometry Non-Euclidean Geometry Thales of Miletus was born in the Asia Minor, where Turkey currently exists around 624 BC. He was the first known Greek philosopher, scientist, and mathematician of his time. Thought it is speculated that he had many discoveries, any writings or findings have not survived the age of time. It is said that he brought back geometry knowledge from Egypt, where he measured pyramid heights using their shadows and is credited with 5 important theorems in elementary geometry: 1. A circle is bisected by any diameter. 2. The base angles of an isosceles triangle are equal. 3. The angles between two intersecting straight lines are equal. 4. Two triangles are congruent if they have two angles and one side equal. 5. An angle in a semicircle is a right angle. Proclus used the findings of Thales to expand and discover his own ideas. Thales died around 547 in his hometown. Pythagoras of Samos was born in Iona 569 BC and died 475. His ideas are very important to modern math, however very little is known about his discoveries. He was influenced by Thales and his student Anaximander, who tickled his interest about math and astronomy and advised him to follow those ideas to Egypt to learn more. He believed that all relations could be deduced to a numerical relationship. Pythagoras is most famous for his theorem for determining the lengths of the sides of a right triangle, but also contributed to other ideas as well, like the sum of the angles of a triangle and constructing figures of a certain area. In his life, he was mostly a philosopher thinking about ethics and beliefs. Hippocrates of Chios lived between 470 BC and 410 BC in Greece. He taught in Athens, was a Pythagorean and worked on classical problems like squaring a circle and duplicating a cube. Through these problems, Hippocrates was able to discover lunes, where a lot of his focus was placed and that a cube could be duplicated if proportional values could be found between a number and its doubled value. He is thought to have written a geometry book, but many think that his ideas were contained in Euclid’s later extensions of Elements. Hippocrates also discovered geometric solutions to quadratic equations and early methods of integration. Exodus of Cnidus lived between 408 and 355 BC in Asia Minor, which is now in present day Turkey. He studied under Archytas who was a follower of Pythagoras in Italy and is thought to have learned about number theory and music. He also studied medicine, astronomy, and set up a school which gained many followers itself. He contributed to Euclid’s Elements by finding a method to compare the ratios of different line segments of irrational lengths in a way cross multiplication is used today. Other contributions made by Exodus were his method of exhaustion and planetary theory on motion. Many believe he inspired Aristotle as well. Euclid of Alexandria lived between 326 and 265 BC in Egypt. He is probably the most famous mathematician in the geometry field due to is works and proofs in The Elements, which have influenced western math over the past 2000 years. It is said through Proclus that Elements organizes many of Exodus’ theorems and perfects other ideas from mathematicians of the time. In the 13 books, he provides definitions and postulates proving ideas about shapes and constructions in terms of geometric shapes, the Euclidean Algorithm and the theory of irrational numbers, to name a few. Archimedes of Syracuse lived between 287 and 212 BC in Sicily, which is now part of Italy. He is thought to have invented a pump still used in many parts of the world today and to have studied under followers of Euclid in Egypt. He would send his friends theorems he discovered with no proofs, some of which these friends would try to pass on as their own; he used this and some false theorems to show that some people will go to great lengths to “discover” new ideas, but have really only pretended to discover the impossible. Though he is known mostly for is inventions, he loved studying geometry and used these results very closely in his discoveries. In addition to the inventions, he perfected integration to find areas and volumes and contributed to a variety of mathematical fields in his time through multiple writings, like Method, On spiral, and On floating bodies. Hipparchus of Rhodes lived between 190 BC and 120 BC beginning his life in Bithynia (now Turkey) and continuing his life in Greece. Much of the known information known about Hipparchus comes from Ptolemy’s Almagest. His early contributions included working on trigonometry tables through chords to solve spherical triangles. His writing, Commentary on Aratus and Eudoxus looked into the works of each man on constellations. He also calculated the length of a year to a high accuracy to 6.5 minutes. Heron lived between 10 and 75 AC in Egypt. He was an important man in geometry and mechanics. It has been difficult to find out exactly when Heron lived and discovered ideas in the math field due to the popularity of his name during this time period. His different books from Metirca deal a lot with geometric ideas; book one looks into area of various regular and non-regular polygons, his famous area formula for triangles using its sides, and the surface of 3D figures too. In books two and three he looks into volumes of 3D figures and splitting volume and area up in ratio to the original or another figure. Cladius Ptolemy was born around 85 AC and died 165 in Egypt. He was an influential astronomer and geographer. His name is a combination of Greek and Roman culture, which suggests that he was also a Roman citizen in addition to his Greek background. The Almagest tied together math and astronomy about how the planets, sun and moon move. He also uses his ideas and theories about the universe in space to compare them to Hipparchus’ observations. Ptolemy also completed work that involved finding the latitudes and longitudes of major places and color, reflection, refraction, using mirrors of various shapes. His work is controversial at times due to inaccuracy and opposing beliefs. Aryabhata I, lived between 476 and 550 in India. He was about 23 years old when he completed Aryabhatiya, which Bhaskara I later commentated on. This book is complied of a summary of Hindu math up until the 6th century written in verses, including 66 rules in mathematics without proof, planetary models, spheres and eclipses. It also looks into algebra, plane and spherical trigonometry, and sums of powers to name a few. He is also known during this time for calculating an accurate estimation for pi. Cladius Ptolemy was born around 85 AC and died 165 in Egypt. He was an influential astronomer and geographer. His name is a combination of Greek and Roman culture, which suggests that he was also a Roman citizen in addition to his Greek background. The Almagest tied together math and astronomy about how the planets, sun and moon move. He also uses his ideas and theories about the universe in space to compare them to Hipparchus’ observations. Ptolemy also completed work that involved finding the latitudes and longitudes of major places and color, reflection, refraction, using mirrors of various shapes. His work is controversial at times due to inaccuracy and opposing beliefs. Hypatia of Alexandria lived between 370 and 415 in Egypt. She was the first woman mathematician to make substantial contributions to the math field. She was the daughter of the mathematician Theon, whom she studied under and lectured on Neo-Platonism. She came to symbolize learning science with paganism, but her work became controversial in the Christian religious community. It is also suggested that she helped her father add to his version of Elements and Ptolemy’s Almagest. Bhaskara I was thought to have lived between 600 and 680 in India. Little is known to about his life, but what is known is through his writings. It is also suggested that the discoveries in his writings were influenced by him being a follower of Aryabhata I. His commentaries about the discoveries of Aryabhata I were on mathematical astronomy in India and approximating the sine function using rational fractions; he also provides an accurate formula to calculate such values with minimal error. Other ideas that Bhaskara I looked into are solution methods for first degree equations, quadratic and cubic equations, and solving equations with more than one unknown variable. Abu Abdulla Al-Battani was born 850 in Mesopotamia (present day Turkey) and died 929 in Iraq; he is also known by other names as well. His family was from a sect that worshipped the stars and focused on the study of astronomy and mathematics, but he was not a big believer in the religion. He was most likely a Muslim based on his full name. He is described in the Fihrist as a leader in geometry, astrology, and astronomy during his time, referring to observations and his own in Ptolemy’s Almagest. He is also noted for discovering and cataloging 489 stars, recalculating the length of a year, and using trig functions rather than geometric methods to solve right triangles. Mohammad Abu'l-Wafa was born in 940 in Khorasan (a region now in modern day Iran) region. He worked as a scientist and mathematician translating early commentaries by Euclid and Diophantus. He also wrote many books relating math to different professions such as business and carpentry. His contributions to geometry include a book on geometric constructions for craftsmen explaining ideas such as right angles and inscribing polygons into circles using a ruler and compass. He is best known for using the tangent function, introducing thee secant and cosecant functions, and devising a more accurate method for calculating tables of sine values. Mohammad Abu'l-Wafa died 998 in Iraq. Omar Khayyam was born in 1038 and died in 1131 in Persia, which is in present day Iran. His way of life was affected by the political events happening in the 1000s; he grew up in an unstable military empire with severe religious conflicts. Though he was enrolled in school, it meant nothing unless the government gave support, which generally did not occur. Despite these difficulties, Khayyam was still able to flourish, writing two books before the age of 25 on algebra and music. He contributed to working on astronomical tables and calendar reform to accurately predict the length of a year. He also discovered a geometric method to solve cubic equations. He also looked at the parallel postulate and his discoveries unintentionally proved properties in non-Euclidean geometries. Muhammad Al-Biruni was born in 973 in Khwarazm, which located currently in Uzbekistan and died 1048 in Ghazna (now in Afghanistan). He grew up and lived his life in a war torn area of the world and began his studies at a young age under the famous mathematician Abu Nasr Mansur. At age 22 he completed a bunch of small works, including one on map projections. One of his most important works was the book titles Shadows relates to many ideas we see in geometry today such as secant and tangent functions, an introduction to polar coordinates, trisecting angles, spherical geometry and other constructions that cannot be done with a ruler and compass. He is most famous for producing India, in which he describes his experience with the culture in the country. René Descartes lived between the years 1596 and 1650 in France and Sweden. His major work includes La Géométrie, which applies algebra to geometry to get what we refer to today as Cartesian geometry. He was educated at the Jesuit College, beginning at age eight. He became a philosopher and also received a law degree, joined a military school and enlisted in the Bavarian Army in 1619. Believing that math was the foundation to many ideas, Descartes researched and discovered in fields such as physics and found some faults in meteorology. Giovanni Saccheri was born September 5th, 1667 in San Remo, Genoa (which is now in Italy). He studied philosophy and theology at a Jesuit College where he later began teaching and became a priest in 1694. His work in mathematics aims at studying two types of definitions: ones that give meaning to a term and ones that give meaning to defend that the term actually exists. His name is remembered in the modern day due to Eugenio Beltrami, which included non-Euclidean geometry and the parallel postulate (John Wallis and Nasir Al-Tusi). He died in 1733 in Milan. Nasir Al-Tusi was born in 1201 in Khorasan, which is in present day Iran and died in 1274 in Kadhimain, near Baghdad, Iraq. He was known by many different names during the time in which he lived. He experienced a life in which the military power of the Mongols swept across the Islamic world and eliminated anyone that threatened their control. As a young boy, he was introduced to topics such as logic, metaphysics, algebra, and geometry. While living in a time of such destruction, Al-Tusi went to Nishapur, where learning was flourishing; it was here that he studied medicine, math, and philosophy. He wrote many commentaries on Greek texts by Euclid, Archimedes, and Ptolemy; as for his own contributions, Nasir Al-Tusi created tables of planetary movements and acknowledged trigonometry as a field on its own away from its applications to astronomy. He also introduced the law of Sines formula and determined coefficients in a binomial expansion to any power using the binomial formula and Pascal’s triangle. Johann Lambert was born in France in 1728 and died in Berlin, Prussia (Germany) in 1777. He went to school at a young age, but left at twelve to join his father’s tailoring business; he continued his studies in his spare time. At twenty, he became a private tutor, which helped him extend his knowledge in math, astronomy, and philosophy. In 1760, Lambert was recommended by Euler to become a Professor at St. Petersburg Academy of Sciences. His study in geometry was through the parallel postulate; he assumed that it was false to come about results that were non-Euclidean. Lambert also used the principals of measurement and extent. He is best known for his rigorous proof showing that π is irrational and developing systematic hyperbolic functions. Johann Gauss lived between 1777 and 1855 in Brunswick and Hanover, which are now parts of Germany. His mathematical intelligence was noticed at a young age; he went to college in Brunswick, studied under Marten Bartles. It was at the Academy that he discovered Bode’s Law, which includes ideas associated with the binomial theorem and the arithmetic geometric mean as well as the prime number theorem. Gauss was also the first to construct a 17 sided regular polygon with a ruler and compass. He wrote books on celestial bodies and related them to differential equations in terms of orbiting and was interested in non-Euclidean geometries and differential geometries. In addition to these ideas, Gauss worked with Wilhelm Weber on physics and magnetism. He accomplishes a lot in his life time and well known for his work. Adrien-Marie Legendre was born in Paris 1753 and died 1833. He was all about his work, which consumed his whole life and liked it to be the center of attention. He came from a wealth family, attended college, and after graduating he focused on his research. He entered a contest offered by the Berlin Academy, which he won and launched his career. Legendre's major work dealt with elliptic integrals as a tool for physics. He also wrote his own version of Elements in 1794, where he simplified proofs to make it more effective to use. He also provided proof that both π and π2 are irrational. He also attempted to prove the parallel postulate, which spanned over 30 years, but failed to do so because he relied on ideas evident by Euclid’s propositions. John Playfair was born in Scotland in 1748. Because his father was a member of the church, he was expected to follow suit; he studied at St. Andrew’s , where he was respected by his professors and ended up delivering lectures in natural philosophy for an ill professor while still in his teens. While trying to gain an academic job, Playfair conducted experiments with others in his field. He was finally accepted to teach at the University of Edinburgh and in 1795 he wrote his own version of Elements. The parallel postulate and theory of proportion caused problems during this time of studying geometry; Playfair came up with an alternative to the parallel postulate, stating “Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line” Which was given by Proclus many years before. He also looked into spherical geometry, geometric solids and sine curves. He was a respected man in the society which he lived until July 1819 when he died in Scotland. Nikolai Lobachevsky was born in Russia in 1792 and died in 1865. Lobachevsky was from a poor family, but graduated from the Gymnasium and attended Kazan University for free where he studied under Martin Bartles, a friend of Gauss. Through Bartles lectures on Euclid, it is suggested that Lobachevsky became interested in the famous fifth postulate. He looked at the postulate from a different perspective that the fifth postulate does not necessarily hold, making Euclid’s version a specific case of general geometry. At times, the government which he lived was not as tolerant of his craft, but under new leadership came more freedoms. He became the dean of Math and Physics at the university. In 1829 Lobachevsky wrote a book on non-Euclidean geometry, which was the first of its kind to appear in print (specifically Hyperbolic). He also taught other subjects like integration, differential equations, calculus, and physics. Georg Friedrich Bernhard Riemann was born in Breselenz, Hanover (Germany) on September 17 1826. He was a good student who showed an interest in math, so the director of his school allowed him to read from his personal collection including Legendre's book on the theory of numbers. He originally began studies in theology at university but later transferred to philosophy and math courses by Gauss and Moritz Stern. Riemann contributed through complex variable and Riemann Surfaces. He lectured on the foundations of geometry and relations to trig functions and also looked at ideas by others like Euler. He died in Selasca, Italy July 1866. Eugenio Beltrami was born in 1835, when his homeland was still part of the Austrian Empire and died in 1900 in what is now part of the Italian peninsula. Eugenio came from an artistic family, where his father was a painter; he himself was also artistic in terms of producing music in addition to his mathematical talent. Beltrami studied mathematics in college, but due to economic hardships he dropped out to get a job. During this time, the Kingdom of Italy was formed, which restricted academic growth. It was also during this time that Beltrami began his studies again and published his first paper in 1862. Eugenio Beltrami was influenced by mathematicians such as Lobachevsky, Gauss and Riemann, which allowed for his contributions to differential geometry though curves and surfaces. He also made concrete connections between the works of these men, like Lobachevsky’s non-Euclidean geometry to Reimann. Felix Klein was born in Prussia (Germany) in 1849 and died in Germany, 1925. He is best known for working with non-Euclidean geometry, connecting geometry and group theory, and function theory. Klein began with the intent of becoming a physicist, but became the assistant to Plücker and worked with him on the foundations of line geometry. He believed that it was possible to consider Euclidean and non-Euclidean geometries as special cases and that if one was consistent, so was the other; this helped relieve the controversy of the subject. Pierre Wantzel was born in Paris, France in 1814. He was a studious man throughout his early life, going to college and teaching math and physics courses at various colleges near Paris. Wantzel is famous for his work dealing with solving equations by radicals. His application to geometry is through proofs first mentioned by Gauss explaining that trisecting an angle and duplicating a cube could not be done simply with a ruler and compass. He was a workaholic who continued to prove new ideas and expand on others until his death on May 21, 1848. Janos Bolyai lived from 1802 and 1860 in parts of Hungary which are now in present day Romania. His father was Farkas Bolyai, who taught math, physics and chemistry at the Calvinist College and was good friends Gauss. His father believed that the best way for his son to develop a sound understanding of math was to focus on his physical well being; Janos was able to identify geometric figures, the sin curve, and taught himself to read at a young age. His father wanted him to study under Gauss to have the best education possible at the time, but Gauss refused the idea; Janos went to college instead and after graduation he entered the Academy of Vienna to study military engineering. He was a studious person, learning to speak nine different languages. In 1820, Bolyai tried to follow in his father’s path, by attempting to replace Euclid’s parallel postulate, but gave up soon after; the work he did discover was on the edge of basic ideas about hyperbolic geometry. He expanded on the idea and wrote up a complete system of his findings. Gauss referred to Janos as a “genius of the first order” for his discovery. David Hilbert lived between 1862 and 1943; he was born in Prussia (now Russia) and also spent time in Germany. He attended the gymnasium in his home town of Königsberg and after graduating he went to the university there as well. He knew Felix Klein as a personal friend and became the math chair at the University of Göttingen, where he taught from 1895 to the end of his career. His first famous piece of work was accomplished in 1888 where he proved the Basis Theorem. David Hilbert had one of the greatest influences on geometry after Euclid; he studied the axioms which led him to propose 21 more and analyzed their findings. His 23 Paris problems have challenged mathematicians past and present for decades now. Hilbert also contributed work to other fields, such as invariants, functional analysis, and integral equations to name a few. The main ideas that have come from calculus are derivatives, and integrals. The ideas that led up to these were developed throughout the 17th century, but there were also discoveries before this time as well. The Greeks used ideas from calculus, including Eudoxus’ method of exhaustion to compute the area and volume under regions and solids, which relates to the early idea of integrals. Archimedes was also the first to find the tangent line to a curve, which relates to derivatives. These ideas were the basis of later mathematicians Isaac Barrow and Johann Bernoulli. It was not until Newton’s time that these methods would be incorporated into the general framework of calculus.

The 17th century was when the idea of the derivative became into talks. Pierre de Fermat developed a method to find the max, min, and tangents to specific curves with a method equivalent to differentiation. In addition to this idea, Fermat also developed a trick to evaluate integrals directly with any power. Sir Isaac Newton would draw ideas from this method in terms of Fermat’s drawing of tangents. The influence of Fermat’s discoveries also influenced both James Gregory and Isaac Barrow to prove the fundamental theorem of calculus.

The two men that gain most of the focus for ideas developed in calculus are Newton (1643-1727) and Leibniz (1646-1716). Newton looked at calculus from and investigative perspective using geometry and physics as his background in terms of movement and magnitude where as Leibniz focused on tangents and leaned towards a metaphysical explanation. Because of these opposite ideas, insight was given to calculus in terms of inverse properties between integrals and derivatives. The fundamental theorem of calculus was built into Newton’s discoveries on the inversion property for Fluxional Calculus in the mid 17th Century.

In Leibniz’s own version of calculus, he argued that the integral was the sum under a given area (ordinates) of infinitely many rectangles. He is also credited with coming up with the modern notation used, like dy and dx for the ordinates and the summation of infinitely many of these rectangles with the current integral symbol . Newton used function notation for derivatives in terms of primes (f(x), f’(x)…).

In terms of change over time, many other mathematicians have adapted the fundamental basics of calculus to look at different aspects in the field and expand on that. Calculus can also be applied to many other fields, like physics, astronomy, chemistry, etc. It has become useful in multiple fields and has allowed for new information to be developed. Modern Algebra Algebra was thought to have originated during the time of the Babylonians about four thousand years ago; priests are said to have used algebra to go along with religious rituals in the culture. Algebra can be broken into sub topics such as elementary algebra which deals with solving equations using arithmetic operations and abstract algebra which looks at the general and abstract ideas behind elementary ideas.

Many different cultures have developed their own system of algebra based on other cultures and ideas before them, like the Egyptians. Babylonian math was some of the most advanced in the time before the discoveries of the Greeks. They developed a number system to guide them to solve algebraic equations using quadratic and cubic expressions in terms of approximations. These numbers systems have been documented through tablets made of clay with etched symbols, many of which were created around 1900 BC. One of the most famous tablets, known as Plimpton 322 gives a table of Pythagorean triples. They were also familiar with adding equals and multiplying both sides of an equation by the same quantity to eliminate factors and fractions.

The Greeks also had their own development of algebra through Euclid and his discoveries in geometry. In book two of Elements, Euclid proposes fourteen propositions with results that are known today to show symbolic algebra and trigonometry. Euclid did not use numbers; instead he used letters and line segments to deduce results through axioms and geometric theorems. Arithmatic operations are also used to solve geometric algebra for the Greeks. Data, which was written by Euclid contains many statements about rules and formulas in algebra. Menaechmus discovered conic sections like ellipses, circles, parabolas, and hyperbolas around 380 BC.

Diophantus is known as the “father of Algebra,” who lived during the Hellenistic period (around 250 BC). His book Arithmetica differs from the math that the Greeks and Babylonians studied; he was concerned with exact answers instead of simple approximations and was the first to use variables/symbols to represent unknown values and operations (syncopated algebra). It is also suggested that Al-Khwarizmi discovered algebra because of his proof of a quadratic with positive roots and for teaching an elementary form of the subject; his work established algebra as a discipline independent from geometry and arithmetic.
More recently in the past 200 years or so, the study of algebra has taken a more abstract approach. Abstract algebra can be studied in many different ways, like group theory, rings, fields, and vector spaces. Mathematicians changed gears in the late 1800s from proving concrete properties for specific cases in algebra to a more general theory to apply the properties to multiple algebraic structures and ideas. David Hilbert, in addition to his other contributions, studied commutativity and rings along with many others. Other well known men associated with modern algebra were Leonhard Euler (modular arithmetic), Carl Friedrich Gauss (cyclic and abeilian multiplicative group), and Joseph LaGrange (permutations and solutions of equations). These developments were presented by Bartel van der Waerden in Moderne Algebra, which was published in 1930 and changed the study of algebra from something about solving equations to a theory of algebraic structures. Number systems are a way to represent numbers in mathematical notation; this could be in terms of a set or series to provide a consistent representation. Number systems have the following basic properties: 1) to represent a useful set of numbers, 2)to give all represented numbers a unique or standard representation, and 3) to reflect algebraic and arithmetic structures of numbers.

The most common numeral system used in the modern day is the Hindu-Arabic system. This was created by Aryabhata I in the 5th century and Brahmagupta added the symbol for zero about a century later; both of these men were from Indian descent. Other number systems still presently used today are the unary or tally system for small numbers, sign-value notation and Roman numerals.

Numeral systems may vary today based on culture and where they are used. Alphabetic numerals, like Greek lettering are used symbolically throughout math to represent operations. In the past, it is possible to expect that each civilization had its own system of numbers and counting, but today in modern time it is more universal in terms of what is used. The history of calculating probabilities originated with the methods developed by Pierre de Fermat and Blaise Pascal in the early 1650s. It is debated whether or not probability theory helped statistics emerge or statistics brought about the study of probability methods. Whatever way you look at it, the two ideas work together to analyze data and random samples. In general, the difference between the two ideas is that statistics uses data to make inferences about it through different hypothesis tests and probability uses random processes that come from the data looking for particular outcomes that could occur.

In terms of the development of probability, Jacob Bernoulli and Abraham de Moivre were the ones to put its usefulness and methods on the mathematical map in the 18th century. Bernoulli proved a version of one of the fundamental theorems used in probability theory today, the Law of Large Numbers; this states that with a large number of trials, the average outcome in a random sample will be close to the expected number you would think to receive and be normally distributed. More developments were made in the 1800s and 1900s as well. Gauss created a method to reduce error in observations, like the method of least squares and the introduction of statistical mechanics by Boltzmann and J. Willard Gibbs to explain properties of gases. Controversy in this area has from how to interpret probability results; frequentism was the dominant method, describing the means are relative to frequency in large trials, but the Bayesian view also came about through how well the evidence supports what is trying to be discovered.

Statistics is related to probability through hypothesis testing of data, which tries to predict the likelihood of an event or outcome of happening again as extreme as it did in the data. Statistics uses probability and statistic inferences to analyze data. This is used today in terms of drug clinical tests, understanding voting outcomes, agriculture research, etc. Statistics is not an individual field in mathematics, but instead a mathematical science because it depends on interpreting data and research to obtain conclusive or inconclusive results.
Discrete mathematics is the study of structures that can often be proven using the integers and natural numbers, more specifically countable sets of numbers that are both finite and infinite. The history of the mathematical subsection has involved a number of different topics and challenging problems, but most of this development has been made in the last century. Discrete mathematics was not a popular field before the 20th century were little was known.

This field of mathematics has been used in recent times in many different ways. In the 1940s during WWII, the need to break enemy codes gave way to the development of cryptology which uses modulo arithmetic to send secret messages without others being able to understand them should the intercept the message. Public-key encryption was not preleased to the masses until Whitfield Diffie found a way to do so in the 1970s. This development was partly due to the production of digital computers. Discrete math is used in computer programming and software development to keep files safe from viruses and hackers.

Logic is also another important part of discrete mathematics because it allows for more straight forward methods of proving ideas to be true or false. In the 1700s, attempts were made to develop formal logic into a symbolic way by Leibniz and Lambert, like the logical equivalencies that we know and use today. George Boole worked on logic through algebraic discoveries and extending the discoveries to studying math. Ernst Schröder published three volumes which summarized the works of Boole, DeMorgan and Pierce to create a reference system for symbolic logic up until the early 1900s.

Through logical arguments made in the mid-1800s, flaws in Euclidean axioms were discovered. The parallel postulate as we know has conflict behind its truth in other geometries and proving certain ideas and was proven independent by Nikolai Lobachevsky in 1826. Some of the ideas proven by Euclid were shown to in fact not be able to be proven by his axioms. Through testing these axioms and proofs with new ideas in logic, David Hilbert developed a new set of geometric axioms that used natural and real numbers by building on the previous work of Moritz Pasch in the late 1800s. This would prove to be a major area of research in the first half of the 20th century where the main areas of study were in set theory and formal logic to look for proofs of consistency in a mathematical world that can be inconsistent.
Units of measurement are one of the earliest tools that were created by humans. In the earliest years known to date, measuring was used for building and constructing housing through determining appropriate size and shape of structures.

In terms of Euclidean geometry, measurement was not done using numerical systems; instead Euclid used an unmarked straight edge and compass to find the size of angles and distance of lines. Angles were not measured in degrees and distance was not found using number systems. Distance was a relative term where a certain line segment of length x was on the real number line. Other lines are then chosen in relation to it. Angles were also measured in a different way than we use today. Instead of using exact measurements in degrees, Euclid used a right angle as his basic unit so that other angles could be referenced from it, like 180 degrees being equal to 2 right angles.

Euclidean measurement also gave way to implying how to calculate area and volume through the distances he measured. His measuring was to prove the different propositions he found to be true. Without giving exact numbers, it showed that the constructions were universal for any line segment and angle in proportion to the original. The proofs that he was able to provide can today be constructed using units and number systems for exact angle measures and lengths of lines. Numerical measurements are also helpful in terms of predicting the size of missing sides and making specific shapes.
The history of geometry dates back to the Egyptians and Babylonians before large advances was made by the Greeks. It can be broken into classic geometry which was founded through Euclid and modern geometry which today uses abstract and complex ideas in relation higher mathematics.

Some of the most important ideas from geometry are the Pythagorean Theorem, the ability to construct figures using only a compass and straightedge, and non-Euclidean geometry.

These first two ideas stem from classic geometry in which we know that Euclid developed and proved through his book Elements. The discovery of the Pythagorean Theorem, which is suggested to have been know by the Egyptians before Pythagoras’ time, is a great tool to determine if a triangle contains a right angle or not and to help determine the side lengths. Construction also goes along with Pythagorean Theorem with the idea of side lengths. Knowing geometric shapes and being able to replicate them is an important contribution to geometry and math in general because it allows visual representation and proof of the work that Euclid and the Greeks were able to accomplish.
In terms of non-Euclidean Geometry was developed around the 18th century due to the many issues of proving Euclid’s parallel postulate. Though many failed at doing so, in particular Omar Khayyam, what was discovered were eventual properties in a new type of geometry. The first men to take on expanding this work were Saccheri, Lambert, and Legendre, but it was not until the 19th century that Gauss, Johann Bolyai, and Lobachevsky really began to see new ideas in the different directions they took on the idea of non-Euclidean Geometry. One of their main ways of doing this was to assume that the parallel postulate was false which provided the opening for Reimann and Beltrami to make their marks in this new field of mathematics. These men introduced ideas like hyperbolic geometry and smooth curves and surfaces. THE END
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