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The Age of Euler

The Age of Euler. Chapter 10 Part 1. Leonhard Euler [1707-1783]. Euler is considered the most prolific mathematician in history. His contemporaries called him “ analysis incarnate .” “ He calculated without effort, just as men breathe or as eagles sustain themselves in the air. ”.

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The Age of Euler

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  1. The Age of Euler Chapter 10 Part 1 The Saga of Mathematics

  2. Leonhard Euler [1707-1783] • Euler is considered the most prolific mathematician in history. • His contemporaries called him “analysis incarnate.” • “He calculated without effort, just as men breathe or as eagles sustain themselves in the air.” The Saga of Mathematics

  3. Leonhard Euler [1707-1783] • Euler was born in Basel, Switzerland, on April 15, 1707. • He received his first schooling from his father Paul, a Calvinist minister, who had studied mathematics under Jacob Bernoulli. • Euler's father wanted his son to follow in his footsteps and, in 1720 at the age of 14, sent him to the University of Basel to prepare for the ministry. The Saga of Mathematics

  4. Leonhard Euler [1707-1783] • At the age of 15, he received his Bachelor’s degree. • In 1723 at the age of 16, Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. • His father demanded he study theology and he did, but eventually through the persuading of Johann Bernoulli, Jacob’s brother, Euler switched to mathematics. The Saga of Mathematics

  5. Leonhard Euler [1707-1783] • Euler completed his studies at the University of Basel in 1726. • He had studied many mathematical works including those by Varignon, Descartes, Newton, Galileo, von Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. • By 1727, he had already published a couple of articles on isochronous curves and submitted an entry for the 1727 Grand Prize of the French Academy on the optimum placement of masts on a ship. The Saga of Mathematics

  6. Leonhard Euler [1707-1783] • Euler did not win but instead received an honorable mention. • He eventually would recoup from this loss by winning the prize 12 times. • What is interesting is that Euler had never been on a ship having come from landlocked Switzerland. • The strength of his work was in the analysis. The Saga of Mathematics

  7. The 18th Century • The rise of scientific and mathematical journals of the preceding century was the quickest way of making new discoveries known. • This outgrowth of the printing revolution of the 15th century accelerated the pace of mathematical and scientific progress by transmitting new ideas in a timely manner. • Similar to the growth of the information age. The Saga of Mathematics

  8. The 18th Century • The 18th century was still an age when no man could consider himself educated without a knowledge of mathematics, for on mathematics all knowledge was based. • The universities were not the principal centers of research. • This nurturing was done by the various royal academies supported by generous rulers, like, Fredrick the Great of Prussia and Catherine the Great of Russia. The Saga of Mathematics

  9. The 18th Century • These academies gave Euler the chance to be the most prolific mathematician of all time. • They were research organizations which paid their leading members to produce scientific research. • Of course, the academicians were paid to produce results but once the rulers got a reasonable return on their investment, Euler, Lagrange, and the others were free to do as they pleased. The Saga of Mathematics

  10. The 18th Century • The rulers of the 18th century let science take its course. • The first practical problem of this age was the control of the seas. • This meant accurate navigation techniques which ultimately requires determining one’s position while out at sea. • This position is determined by observing the heavens. The Saga of Mathematics

  11. The 18th Century • After Newton’s universal law suggested that the position of the planets and the phases of the Moon could be calculated for centuries in advance, those wanting to rule the seas started number crunching. • The Moon offers a particularly difficult problem involving three bodies attracting one another; the Moon, the Earth and the Sun. • Euler was the first to derive an approximate solution. The Saga of Mathematics

  12. Leonhard Euler [1707-1783] • Euler eventually obtained royal appointments in several European courts including Russia and Germany (under Frederick the Great). • Two of Euler’s friends, Daniel and Nicholas Bernoulli, encouraged Catherine I (wife of Peter the Great) to appoint him a position in the medical section at St. Petersburg. • Euler quickly attended lectures on medicine at Basel in hopes of obtaining the post, which he received in 1727. The Saga of Mathematics

  13. Leonhard Euler [1707-1783] • Even in physiology, Euler could not keep away from mathematics. • The physiology of the ear suggested an investigation of sound, which in turn led to the propagation of waves. • Euler eventually wrote an article on acoustics, which went on to become a classic. • Quantity as well as quality characterized Euler’s work. The Saga of Mathematics

  14. Leonhard Euler [1707-1783] • Upon Nicholas Bernoulli’s death, Euler was appointed as head of the Natural Philosophy department. • In 1733, Daniel Bernoulli returned to Switzerland and Euler, at the age of 26, was appointed to senior chair of mathematics. • The publication of many articles and his book Mechanica (1736-37) – a two volume book on mechanics – started him on the way to major mathematical work. The Saga of Mathematics

  15. Euler’s Mechanica (1736) • First textbook in which Newton’s dynamics of the mass point was developed with analytical methods. • Followed by the Theoria motus corporum solidorum seu rigidorum (1765) in which the mechanics of solid bodies was similarly treated. • The later contains the “Eulerian” equations for a body rotating about a point. The Saga of Mathematics

  16. Euler and the Atheist A Famous Tale • Catherine the Great had Denis Diderot, a French philosopher and editor of the great French Encyclopédie, visit her Court. • Diderot an atheist tried to convert the courtiers to atheism. • Fed up with Diderot, Catherine asked Euler to puzzle him. • Diderot was informed that a learned mathematician was in possession of an algebraic proof of the existence of God. The Saga of Mathematics

  17. Euler and the Atheist • Diderot consented to hear it even though he knew nothing about mathematics. • As the story goes, Euler approached Diderot and said, “Monsieur, donc Dieu existe; répondez!” • That is, “Sir, , hence God exists; reply!” The Saga of Mathematics

  18. Euler and the Atheist • This sounded like sense to Diderot. • He was humiliated by the uncontrolled laughter. • Diderot asked permission to return to France at once, which was granted. • Of course, Euler’s argument was nonsense but Diderot didn’t see it. • Euler would eventually meet his match in arguments with Voltaire. The Saga of Mathematics

  19. Leonhard Euler [1707-1783] • Euler had a phenomenal memory. • As a boy, Euler memorized Virgil’s Aeneid and could recite it flawlessly the rest of his life. • Euler not only memorized the first 100 prime numbers but also their squares, cubes, fourth, fifth and sixth powers! • He could also perform difficult calculations mentally, some of which required him to retain in his head 50 places of accuracy. The Saga of Mathematics

  20. Leonhard Euler [1707-1783] • Euler’s constant outflow of ideas is legendary. • It is said that he would write a mathematical paper in the half hour between the first and second calls for dinner. • He published three monumental works on analysis, and also wrote on algebra, arithmetic, mechanics, music, chemistry, and astronomy. The Saga of Mathematics

  21. Leonhard Euler [1707-1783] • In 1741, Euler was invited by Frederick the Great of Prussia to come to Berlin to teach and do research. • In Berlin, Euler published his Introductio in Analysin infinitorum (1748). • This was followed by Institutiones calculi differentialis (1755) and the three volume Institutiones calculi integralis (1768-74). • Instantly became classics. The Saga of Mathematics

  22. Euler’s Analysis Infinitorum • Divided into two parts: • Algebra, theory of equations and trigonometry • Analytical geometry • It contains the expansion of various functions in series and the summation of certain series. The Saga of Mathematics

  23. ei + 1 = 0 Euler’s Analysis Infinitorum • He pointed out that an infinite series cannot be safely added unless it is convergent. • Although he recognized this necessity for dealing with series, he often failed to observe it in much of his own work. • He introduced the current abbreviations for the trigonometric functions, and showed that ei = cos  + i sin . The Saga of Mathematics

  24. Euler’s Analysis Infinitorum • Euler showed that the general equation of second degree Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents the various conic sections. • He extended the application of analytical geometry to three dimensions, where he found general forms for the equations of different solids. • A circle centered at the origin is given by the equation x2 + y2 = r2 and a sphere centered at the origin is given by x2 + y2 + z2 = r2. The Saga of Mathematics

  25. Euler’s Institutiones calculi integralis • A thorough investigation of integrals. • It includes Taylor’s theorem with many applications. • The Beta and Gamma functions were invented by Euler and he uses them as examples of integration. • As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work. The Saga of Mathematics

  26. Leonhard Euler [1707-1783] • Although he lost the sight in one eye in 1735 and the other eye in 1766, nothing could interrupt his enormous productivity. • In 1770 Euler published his Vollständige Anleitung zur Algebra. • A French translation with numerous and valuable additions by Lagrange appeared in 1774. • In this text, Euler proves xn + yn = zn is impossible for integers x, y, z, n=3 and n=4. (Fermat’s Last Theorem) The Saga of Mathematics

  27. Leonhard Euler [1707-1783] • In 1744 appeared Euler’s Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. • He includes solutions to the classic problems on isoperimetrical curves, the brachistochrone in a resisting medium, and the theory of geodesics. • It was this that lead him to the calculus of variations, a sort of generalization of calculus. The Saga of Mathematics

  28. Other works by Euler • His most important works on astronomy in which he attacked the problem of three bodies are: • Theoria Motuum Planetarum et Cometarum (1744). • Theoria Motus Lunaris (1753) • Theoria Motuum Lunae (1772) • His three volume work on optics Dioptrica (1769-71). The Saga of Mathematics

  29. Other works by Euler • In 1739 appeared his new theory of music Tentamen novae theoriae musicae which, it is said, was too musical for mathematicians and too mathematical for musicians. • Lettres a une princess d'Allemagne sur divers sujets de physique & de philosophie (1760-61) were composed to give lessons in physics, mechanics, optics, astronomy and sound. The Saga of Mathematics

  30. Euler’s Letters to a German Princess • During Euler’s stay in Berlin (1741-66), he was asked to provide some tutoring in Natural Philosophy (elementary science) to Princess d'Anhalt Dessau, a niece of Frederick the Great. • These lectures were published in several volumes entitled Letters to a German Princess (1760-61), and for half a century they remained a standard treatise on the subject. The Saga of Mathematics

  31. Euler’s Letters to a German Princess • They became immensely popular and were circulated in seven languages. • William Dunham says the they are one of history’s finest example of “popular science.” • What we call Venn diagrams first appears in Euler’s Letters. • Venn himself first called them "Eulerian Circles", but then somehow managed to get them called Venn Diagrams. The Saga of Mathematics

  32. Leonhard Euler [1707-1783] • Many other results of Euler can be found in his smaller papers. • Some of the better known results are: • Euler’s Polyhedron Formula: V – E + F = 2. • The Euler Line of a Triangle. • Euler’s constant  = 0.577215664901532…. • Euler's theorem (also known as the Fermat-Euler theorem). • Euler’s pentagonal formula for partitions. • Eulerian graphs The Saga of Mathematics

  33. Leonhard Euler [1707-1783] • Euler was in a sense the creator of modern mathematical expression. • In terms of mathematical notation, Euler was the person who gave us: •  for pi • ifor 1 • y for the change in y • f(x) for a function •  for summation The Saga of Mathematics

  34. Leonhard Euler [1707-1783] • To get an idea of the magnitude of Euler’s work it is worth noting that: • Euler wrote more than 500 books and papers during his lifetime – about 800 pages per year. • After Euler’s death, it took over forty years for the backlog of his work to appear in print. • Approximately 400 more publications. The Saga of Mathematics

  35. Leonhard Euler [1707-1783] • He published so many mathematics articles that his collected works Opera Omnia already fill 73 large volumes – tens of thousands of pages – with more volumes still to come. • More than half of the volumes of Opera Omnia deal with applications of mathematics – acoustics, engineering, mechanics, astronomy, and optical devices (telescopes and microscopes). The Saga of Mathematics

  36. Leonhard Euler [1707-1783] • His publications account for one-third of all the technical articles published in 18th century Europe. • He lost his sight sometime after 1766, yet he continued his research at his usual energetic pace while his students wrote it down. • So, what areas of math did he enrich and expand? The Saga of Mathematics

  37. Leonhard Euler [1707-1783] • The question is what field of math did he not enrich and expand! • Not only did he contribute substantially to • Calculus • Geometry • Algebra • Mechanics • and Number Theory • He invented several fields. The Saga of Mathematics

  38. Leonhard Euler [1707-1783] • Euler was the father of thirteen children (all but five died very young) and still found time to become the father of an important branch of mathematics, known today as graph theory. • Important in such fields as computer science, networking, operations research, physics and chemistry. • Euler became the father of graph theory after solving the “Seven Bridges of Königsberg” problem. The Saga of Mathematics

  39. The Bridges of Königsberg Problem • In 1736, Euler published his solution to the problem known as the Seven Bridges of Königsberg in a paper Solutio problematis ad geometriam situs pertinentis. • This paper is considered to be the earliest application of graph theory or topology. • It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. The Saga of Mathematics

  40. The Seven Bridges of Königsberg A D B C The Saga of Mathematics

  41. The Bridges of Königsberg Problem • The Problem: Find a route that crosses each bridge exactly once and returns to where it starts. • Euler observed that it could not be done! • Each landmass has an odd number of bridges. • A traveler departing, returning, departing, etc. an odd number of times would wind up departing on the last bridge, making it impossible to return to the point of origin. The Saga of Mathematics

  42. The Bridges of Königsberg Problem • Let’s consider this gem of thinking one more time. • Number the bridges contiguous with landmass A, 1, 2, and 3. • If one starts the trip by departing A on bridge #1, they must return on bridge #2 or #3, leaving only one more bridge. • They must depart on the bridge not yet traveled on – and that makes all the difference! • You cannot end your trip on landmass A. The Saga of Mathematics

  43. The Bridges of Königsberg Problem • Observe that the sizes of the land masses as well as the lengths and shapes of the bridges are irrelevant. • Thus, you can redraw the diagram above with the landmasses as dots and the bridges as lines. • See the Figure. The Saga of Mathematics

  44. Leonhard Euler [1707-1783] • Notice the irrelevance of the weird shapes of the bridges meeting at B. • The lengths of the lines and the precise locations of the dots are also unimportant. • Euler considered this problem in the context of Leibniz’s desire for a type of geometry that doesn’t involve the concept of a metric such as length or distance. • This is topology or rubber-sheet geometry –The problem is the same if you draw it on rubber and stretch it. The Saga of Mathematics

  45. Euler’s letter to Giovanni Marinoni • “This question is so banal, but seemed to me worthy of attention in that neither geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind whether such a round trip is possible.” The Saga of Mathematics

  46. §1: Graphs in Graph Theory • Today the problem is solved by looking at a graph, or a network, with points representing the land masses and lines representing the bridges. • We define a graph as follows: • A graph G is a collection of dots (called vertices), and a collection of lines (called edges), each line rendering a pair of vertices adjacent. • That is, the edge links the two vertices. The Saga of Mathematics

  47. Visual Representationof a Simple Graph Definition of a Graph • A graphG=(V,E)consists of: • a set V = V(G) of vertices ornodes, and • a set E = E(G) of edges: unordered pairs of distinct elements u,vV. The Saga of Mathematics

  48. ME NH VT NY MA PA CT RI NJ Example of a Graph • Let V be the set of states in the north eastern part of the U.S.: • V={ME, NH, VT, MA, RI, CT, NY, NJ, PA} • Let E={{u,v}|u adjoins v} ={{ME,NH},{NH,VT},{NH,MA},{VT,MA},{VT,NY},{NY,MA},{NY,CT},{NY,NJ},{NY,PA},{MA,RI},{MA,CT},{CT,RI},{NJ,PA}} The Saga of Mathematics

  49. ME NH VT NY MA PA CT RI NJ Example of a Graph (continued) • The specific layout, or representation, of the graph doesn’t matter, as long as the adjacencies and non-adjacencies are preserved. • CT is not that close to NJ! • Note: There is an edge between two vertices if the share a border. The Saga of Mathematics

  50. Directed Graphs • A directed graph or digraphD = (V,A) consists of a set V of nodes together with a set A of ordered pairs of distinct nodes in V called directed edges or arcs. • E.g.: V = species in an ecosystem,A={(x,y) | x preys on y} A food web The Saga of Mathematics

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