Abstract—The transition to calculus was a remarkable period in the history of mathematics and witnessed great advancements in this field. The great minds of the 17th through the 19 Centuries worked rigorously on the theory and the application of calculus. One theory started another one, and details needed justifications. In turn, this started a new mathematical era developing the incredible field of calculus on the hands of the most intelligent people of ancient times. In this paper, we focus on an amazing mathematician who excelled in pure mathematics despite his physical inability of total blindness. This mathematician is Leonard Euler.
Index Terms—Leonard Euler, Euler Characteristic, Seven Bridges of Konigsberg, Zeta Function
Introduction
The invention of calculus started in the second half of the 17th Century. The few preceding centuries, known as the Renaissance period, marked a time of prosperity in different areas throughout Europe. Different philosophies emerged which resulted in a new form of mindset. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artists and scientists such as Leonardo da Vinci. It is no surprise that revolutionary work in science and
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mathematics was soon to emerge at that time. Calculus was very much anticipated as important advancements in mathematics were developed during the Renaissance. Calculus was a rich subject among mathematicians of the 17th Century through the 19th Century. The idea of calculus stemmed from the need to understand motion and dynamic change such as the orbits of the planets and motion of fluids, motion of falling items, and the force of gravity on moving items. The field of calculus indeed focuses on limits, functions, derivatives, integrals, and infinite series. In ancient history, calculus was referred to as infinitesimal calculus [3]. New developments and theories were constantly taking shape in this field thus marking a period where the most intelligent mathematicians laid foundation for the incredible field of calculus. There are numerous notable mathematicians of this era. Jean Le Rond d’Alembert (1717-1783) developed fundamental papers on partial differential equations. Daniel Bernoulli (1700-1782) whose most important work related the basic properties of fluid flow, pressure, density and velocity by what is now known as the Bernoulli’s principle. Jean Baptiste Josheph Fourier (1768 – 1830) who is known by the Fourier Series which is named after him to give him credit on his work on trigonometric series. The Fourier series is a way to represent a wave-like function as the sum of simple sine waves [2]. Joseph-Louis Lagrange (1736-1813) excelled in all fields of analysis and number theory and analytical and celestial mechanics. Augustin Louis Cauchy (1789-1857) made important contributions in the study of analysis and theory of permutation groups. He also researched the divergence and convergence of infinite series. He studied differential equations, determinants, probability and mathematical physics. Many of the theorems known today are named after him such as the Cauchy general theorem and the Cauchy sequence [1]. There are many more mathematicians that helped build the field of calculus. We want to focus on an outstanding one in this paper because of his astonishing mathematical ability and ambition to work despite his physical ability. This mathematician who left a mark in mathematics is Leonard Euler. Leonard Euler It is with no doubt that anyone who studied mathematics came across the name “Euler” at some point.
Euler was one of the mathematical giants of the 18th Century. Leonard Euler (1707-1783) was born in Basel, Switzerland. His father was a Lutheran minister and wanted him to follow his path. Euler’s interest was different however, he was a natural mathematician. Johann Bernoulli helped Euler pursue his path by convincing his father of his mathematical abilities. Bernoulli became Euler’s teacher at the St. Petersburg Academy of Science. Euler’s personal life was more on the tragic side. He married and had 13 children, but only 5 survived their infancy. It is said that Euler made some of his greatest discoveries while holding his baby
[1]. Fig. 1: Leonard Euler [3] Euler had a genius mind and amazing memory skills. Historic papers reveal that Euler produced on average one mathematical paper every week as he compensated for it with his mental calculation skills and photographic memory. For example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. [3]. Euler joined the Berlin Academy of Science in 1741 in response to an invitation from Frederick the Great. He stayed there for 25 years, and wrote over 200 articles. When he returned to Russia in 1766, Euler lost sight of his right eye, he was only 31 years old. Soon after that he became entirely blind. However this did not inhibit his mathematical ability and discoveries. His remarkable memory certainly served him well and so he was able to continue his work on optics, algebra, and motion. When Euler was 58 years old, he had produced almost half of his work despite the total blindness. After his death in 1783, St. Petersburg Academy continued to publish his work for nearly 50 additional years. During his life, Euler made great contributions in modern analytic geometry, trigonometry, calculus, and number theory. He integrated Leibniz work on differential calculus, and Newton’s method of fluxions. Euler advanced integral calculus to a higher degree of perfection. He developed the theory of trigonometric and logarithmic functions and reduced analytical operations to a great simplicity. Euler left a mark on nearly all parts of pure mathematics. Perhaps Euler earned his reputation in 1735 when he worked on the calculation of infinite sums. The problem he worked on was finding the precise sum of the reciprocals of the squares of all natural numbers to infinity (a zeta function using a zeta constant of 2). The problem represented mathematically is 1⁄1^2 +1⁄2^2 +1⁄3^2 +⋯+1⁄n^2 , where n is a natural number. Euler’s friend Daniel Bernoulli estimated the sum to be about 1 3⁄5. Euler tried and was able to come up with the exact and unexpected result of π2⁄6. His method proved to be superior in which he also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire Reimann’s investigation of complex zeta functions [3]. It is worth noting also that in 1735, Euler tackled and solved a problem which was the essence of graph theory. He investigated the known Seven Bridges of Konigsberg Problem, which has perplexed scholars for many years. The city of Konigsberg in Prussia, which is present day Kaliningrad in Russia, was set on both sides of the Pregel River, which included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a route through the city that would cross each bridge once and only once. This is basically represented in graph theory by a planar graph made up of four vertices or nodes (land mass) and 7 edges representing the bridges as depicted in the figure below: Fig. 2: Seven Bridges of Konigsberg Problem [3] Euler proved that there is no solution to this problem because there is an odd number of bridges connecting the nodes. However, if there is an even number of bridges connecting the nodes, a solution would be possible. This laid the foundation of graph theory and prefigured the idea of topology. Another important calculation of Euler used in graph theory and topology is the Euler Characteristic, which, for any surface of polyhedral, the number of vertices minus the number of edges plus the number of faces will always be equal to 2. Fig. 3: The Euler Characteristic [3] It is very helpful to us today that Euler was a plentiful writer, he wrote nearly 900 books. He made many other important findings in mathematics. Euler is regarded as one of the greatest mathematicians of all time. In addition to his discoveries, most of the upper level mathematics notation we use today were created, popularized, or standardized by Euler. For example, the notations we use today that we know from Euler are the base natural algorithm (e), the symbol for imaginary unit equal to the square root of -1 (i), the notation for a function as f(x), the symbol sigma which represents the sum of a set of numbers (Ʃ), the consensus that letters such as a, b, c represent constants as in the sides of a triangle for instance versus letters such as x, y, z represent variables or unknowns, the abbreviations for the trigonometric functions sin, cos, tan, cot, sec, csc, and finally the symbol for the ratio of a circle’s circumference to its diameter represented by π. Conclusion The transition to calculus began with the advancement of mathematics. Important findings made by the intelligent mathematicians of the 16th and 17th Century made the study of calculus a possibility and a reality. One of the greatest mathematicians of all time is Leonard Euler. His name is guaranteed to appear in almost every single upper level mathematics book today. His contributions in pure mathematics were so important that they are highlighted by the Mathematical Association of America and honored by mathematicians around the world. Euler’s legacy has been enormous in terms of shaping the modern field of mathematics and engineering. Without Euler’s work, we wouldn’t have reached a level in applied mathematics and physics that we have today.
From the fourteenth to the seventeenth century the Renaissance transformed European culture and society. Many classical texts resurfaced and new scientific techniques arose. To many, Leonardo da Vinci is one of the most important figures in Renaissance history. He was given the name “Renaissance Man” because of his large role and impact. He had a large list of interests that spanned from science, art, anatomy, architecture, and mathematics. All of which were fundamental components that shaped the Renaissance era into what we know it as today.
Ball, Rouse. “Sir Isaac Newton.” A Short Account of the History of Mathematics. 4th ed. Print.
The first generation of Bernoulli mathematicians include brothers Jacob I(James, Jacques) (1654-1705), Nicolaus (1662-1716), and Johann I(John, Jean) (1667-1748), second generation are brothers Daniel I (1700-1782), Johann II(1710-1790), and their cousin Nicolaus II (1687-1759), and the third generation are brothers Johann III(1746-1807) and Jacob II(1759-1789). It would be exhausting to discuss the accomplishments of all the Bernoulli mathematicians, so our focus will be on the brothers Jacob I and Johann I, who contributed a substantial amount to the fields of mathematics we know today as elementary calculus and the theory of probability.
It is well known that in the past, Renaissance artists received their training in an atmosphere of artists and mathematicians studying and learning together (Emmer 2). People also suggest that the art of the future will depend on new technologies, computer graphics in particular (Emmer 1). There are many mathematical advantages to using computer graphics. They can help to visualize phenomena and to understand how to solve new problems (Emmer 2). “The use of ‘visual computers’ gives rise to new challenges for mathematicians. At the same time, computer graphics might in the future be the unifying language between art and science” (Emmer 3).
The argument in this paper that even though the onus of the discovery of calculus lies with Isaac Newton, the credit goes to Leibniz for the simple fact that he was the one who published his works first. Appending to this is the fact that the calculus wars that ensue was merely and egotistic battle between humans succumbing to their bare primal instincts. To commence, a brief historical explanation must be given about both individuals prior to stating their cases.
...st important scientists in history. It is said that they both shaped the sciences and mathematics that we use and study today. Euclid’s postulates and Archimedes’ calculus are both important fundamentals and tools in mathematics, while discoveries, such Archimedes’ method of using water to measure the volume of an irregularly shaped object, helped shaped all of today’s physics and scientific principles. It is for these reasons that they are remembered for their contributions to the world of mathematics and sciences today, and will continue to be remembered for years to come.
Number theory has to do with numbers of course, but it goes in depth and discusses how numbers relate to one another. Euler committed much of his time to number theory concerning topics such as the Pell equation, Fermat’s Last Theorem, perfect numbers, and the quadratic reciprocity law. Euler developed a theorem that proved Fermat’s theorem and created a deep understanding of Fermat’s theorem by doing so. Euler did not only do work concerning theorems made by other mathematicians, he developed identities and equations himself that are still in use today. For example, Euler’s identity, an equation that concerns many different fields of math. Euler’s formula is another equation that works in pair with his identity equation. These equations are considered beautiful to many modern mathematicians and have not been forgotten. The equations that Euler created, helped make a correlation between different topics and helped many different mathematicians. Euler also introduced new ways to solve quartic equations, and different ways to apply calculus to real life problems. The list goes on, with Euler’s development of Euler’s circle, Euler’s Characteristic, and even proofs. Euler also discussed the problem known as Seven Bridges of Konigsberg. He provided a solution to this problem which led to a theory called graph theory. Euler contributed much more than what was listed, but these are some of the greatest recognized works he
Many years ago humans discovered that with the use of mathematical calculations many things can be calculated in the world and even the universe. Mathematics consists of many different operations. The most important that is used by mathematicians, scientists and engineers is the derivative. Derivatives can help make calculations of anything with respect to another event or thing. Derivatives are mostly common when used with respect to time. This is a very important tool in this revolutionary world. With derivatives we can calculate the rate of change of anything with respect to time. This way we can have a sort of knowledge of upcoming events, and the different behaviors events can present. For example the population growth can be estimated applying derivatives. Not only population growth, but for example when dealing with plagues there can be certain control. An other example can be with diseases, taking all this events together a conclusion can be made.
Leonhard Euler was a brilliant Swiss mathematician and physicist, living between 1707 and 1783. Euler had a phenomenal memory, so much so that he continued to contribute to the field of mathematics even after he went blind in 1766. He was the most productive mathematical writer of all time, publishing over 800 papers. Euler’s dedication towards the subject intrigued me and motivated me to choose a topic related to Euler himself. Amidst his many contributions, I came across e. After further research, I soon learned the multiple applications of the number, and its significance to math. I chose to study the topic of e because I wanted to learn the many ways e can be represented and how it impacts our lives, as well as to share my findings with my peers.
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
The Scientific Revolution was sparked through Nicolaus Copernicusí unique use of mathematics. His methods developed from Greek astr...
Carl Friedrich Gauss was born April 30, 1777 in Brunswick, Germany to a stern father and a loving mother. At a young age, his mother sensed how intelligent her son was and insisted on sending him to school to develop even though his dad displayed much resistance to the idea. The first test of Gauss’ brilliance was at age ten in his arithmetic class when the teacher asked the students to find the sum of all whole numbers 1 to 100. In his mind, Gauss was able to connect that 1+100=101, 2+99=101, and so on, deducing that all 50 pairs of numbers would equal 101. By this logic all Gauss had to do was multiply 50 by 101 and get his answer of 5,050. Gauss was bound to the mathematics field when at the age of 14, Gauss met the Duke of Brunswick. The duke was so astounded by Gauss’ photographic memory that he financially supported him through his studies at Caroline College and other universities afterwards. A major feat that Gauss had while he was enrolled college helped him decide that he wanted to focus on studying mathematics as opposed to languages. Besides his life of math, Gauss also had six children, three with Johanna Osthoff and three with his first deceased wife’s best fri...
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...