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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

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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