Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
Calculus
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Modern calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science and economics. Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has been called "the calculus of infinitesimals", or "infinitesimal calculus"; the term calculus is used for naming specific methods of calculation or notation as well as some theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, process calculus.
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz but elements of it appeared in ancient Greece in China and the Middle East, still again in medieval Europe and in India. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus, but the formulas are simple instructions, with no indication as to method, some of them lack major components. From the age of Greek mathematics, Eudoxus used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes developed this idea further, inventing heuristics which resemble the methods of integral calculus; the method of exhaustion was discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would be called Cavalieri's principle to find the volume of a sphere.
In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, turn calculus into the great problem-solving tool we have today". In Europe, the foundational work was a treatise written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections.
The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, was only rediscovered in the early 20th century, so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, the infinitesimal quantities he introduced were disreputable at first; the formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term; the combination was achieved by John Wallis, Isaac Barrow, James Gregory, the latter two proving the second fundamental theorem of calculus around 1670. The product rule and chain rule, the notions of higher derivatives and Taylor series, of analytic functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics.
In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, it was clear that he understood the principles of the Taylor series, he did not publish all these discoveries, at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, accused of plagiarism by Newton, he is now regarded as an independ
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Francis for the Mathematical Association of America; the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals; the American Mathematical Monthly is the most read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997-2010 are available online; the MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the American Mathematical Monthly. 2017-: Susan Colley 2012-2016: Scott T. Chapman 2007-2011: Daniel J. Velleman 2002-2006: Bruce Palka 1997-2001: Roger A.
Horn 1992-1996: John H. Ewing 1987-1991: Herbert S. Wilf 1982-1986: Paul Richard Halmos 1978-1981: Ralph Philip Boas, Jr. 1977-1978: Alex Rosenberg and Ralph Philip Boas Jr. 1974-1976: Alex Rosenberg 1969-1973: Harley Flanders 1967-1968: Robert Abraham Rosenbaum 1962-1966: Frederick Arthur Ficken 1957-1961: Ralph Duncan James 1952-1956: Carl Barnett Allendoerfer 1947-1951: Carroll Vincent Newsom 1942-1946: Lester Randolph Ford 1937-1941: Elton James Moulton 1932-1936: Walter Buckingham Carver 1927-1931: William Henry Bussey 1923-1926: Walter Burton Ford 1922: Albert Arnold Bennett 1919-1921: Raymond Clare Archibald 1918: Robert Daniel Carmichael 1916-1917: Herbert Ellsworth Slaught 1914-1915: Board of editors: C. H. Ashton, R. P. Baker, W. C. Brenke, W. H. Bussey, W. DeW. Cairns, Florian Cajori, R. D. Carmichael, D. R. Curtiss, I. M. DeLong, B. F. Finkel, E. R. Hedrick, L. C. Karpinski, G. A. Miller, W. H. Roever, H. E. Slaught 1913: Herbert Ellsworth Slaught 1909-1912: Benjamin Franklin Finkel, Herbert Ellsworth Slaught, George Abram Miller 1907-1908: Benjamin Franklin Finkel, Herbert Ellsworth Slaught 1905-1906: Benjamin Franklin Finkel, Leonard Eugene Dickson, Oliver Edmunds Glenn 1904: Benjamin Franklin Finkel, Leonard Eugene Dickson, Saul Epsteen 1903: Benjamin Franklin Finkel, Leonard Eugene Dickson 1894-1902: Benjamin Franklin Finkel, John Marvin Colaw Mathematics Magazine Notices of the American Mathematical Society, another "most read mathematics journal in the world" American Mathematical Monthly homepage Archive of tables of contents with article summaries Mathematical Association of America American Mathematical Monthly on JSTOR The American mathematical monthly, hathitrust
Strasbourg
Strasbourg is the capital and largest city of the Grand Est region of France and is the official seat of the European Parliament. Located at the border with Germany in the historic region of Alsace, it is the capital of the Bas-Rhin department. In 2016, the city proper had 279,284 inhabitants and both the Eurométropole de Strasbourg and the Arrondissement of Strasbourg had 491,409 inhabitants. Strasbourg's metropolitan area had a population of 785,839 in 2015, making it the ninth largest metro area in France and home to 13% of the Grand Est region's inhabitants; the transnational Eurodistrict Strasbourg-Ortenau had a population of 915,000 inhabitants in 2014. Strasbourg is one of the de facto capitals of the European Union, as it is the seat of several European institutions, such as the Council of Europe and the Eurocorps, as well as the European Parliament and the European Ombudsman of the European Union; the city is the seat of the Central Commission for Navigation on the Rhine and the International Institute of Human Rights.
Strasbourg's historic city centre, the Grande Île, was classified a World Heritage Site by UNESCO in 1988, the first time such an honour was placed on an entire city centre. Strasbourg is immersed in Franco-German culture and although violently disputed throughout history, has been a cultural bridge between France and Germany for centuries through the University of Strasbourg the second largest in France, the coexistence of Catholic and Protestant culture, it is home to the largest Islamic place of worship in France, the Strasbourg Grand Mosque. Economically, Strasbourg is an important centre of manufacturing and engineering, as well as a hub of road and river transportation; the port of Strasbourg is the second largest on the Rhine after Germany. Before the 5th century, the city was known as Argantorati, a Celtic Gaulish name Latinized first as Argentorate, as Argentoratum; that Gaulish name is a compound of -rati, the Gaulish word for fortified enclosures, cognate to the Old Irish ráth, arganto-, the Gaulish word for silver, but any precious metal gold, suggesting either a fortified enclosure located by a river gold mining site, or hoarding gold mined in the nearby rivers.
After the 5th century, the city became known by a different name Gallicized as Strasbourg. That name is of Germanic origin and means "Town of roads"; the modern Stras- is cognate to the German Straße and English street, all of which are derived from Latin strata, while -bourg is cognate to the German Burg and English borough, all of which are derived from Proto-Germanic *burgz. Gregory of Tours was the first to mention the name change: in the tenth book of his History of the Franks written shortly after 590 he said that Egidius, Bishop of Reims, accused of plotting against King Childebert II of Austrasia in favor of his uncle King Chilperic I of Neustria, was tried by a synod of Austrasian bishops in Metz in November 590, found guilty and removed from the priesthood taken "ad Argentoratensem urbem, quam nunc Strateburgum vocant", where he was exiled. Strasbourg is situated at the eastern border of France with Germany; this border is formed by the Rhine, which forms the eastern border of the modern city, facing across the river to the German town Kehl.
The historic core of Strasbourg however lies on the Grande Île in the river Ill, which here flows parallel to, 4 kilometres from, the Rhine. The natural courses of the two rivers join some distance downstream of Strasbourg, although several artificial waterways now connect them within the city; the city lies in the Upper Rhine Plain, at between 132 metres and 151 metres above sea level, with the upland areas of the Vosges Mountains some 20 km to the west and the Black Forest 25 km to the east. This section of the Rhine valley is a major axis of north–south travel, with river traffic on the Rhine itself, major roads and railways paralleling it on both banks; the city is some 397 kilometres east of Paris. The mouth of the Rhine lies 450 kilometres to the north, or 650 kilometres as the river flows, whilst the head of navigation in Basel is some 100 kilometres to the south, or 150 kilometres by river. In spite of its position far inland, Strasbourg's climate is classified as oceanic, but a "semicontinental" climate with some degree of maritime influence in relation to the mild patterns of Western and Southern France.
The city has warm sunny summers and cool, overcast winters. Precipitation is elevated from mid-spring to the end of summer, but remains constant throughout the year, totaling 631.4 mm annually. On average, snow falls 30 days per year; the highest temperature recorded was 38.5 °C in August 2003, during the 2003 European heat wave. The lowest temperature eve
Florian Cajori
Florian Cajori was a Swiss-American historian of mathematics. Florian Cajori was born in Switzerland, was the son of Georg Cajori and Catherine Camenisch, he attended schools first in Zillis and in Chur. In 1875, Florian Cajori emigrated to the United States at the age of sixteen, attended the State Normal school in Whitewater, Wisconsin. After graduating in 1878, he taught in a country school, later began studying mathematics at University of Wisconsin–Madison. In 1883, Cajori received both his bachelor's and master's degrees from the University of Wisconsin–Madison attended Johns Hopkins University for 8 months in between degrees, he taught for a few years at Tulane University, before being appointed as professor of applied mathematics there in 1887. He was driven north by tuberculosis, he founded the Colorado College Scientific Society and taught at Colorado College where he held the chair in physics from 1889 to 1898 and the chair in mathematics from 1898 to 1918. He was the position Dean of the engineering department.
While at Colorado, he received his doctorate from Tulane in 1894, married Elizabeth G. Edwards in 1890 and had one son. Cajori's A History of Mathematics was the first popular presentation of the history of mathematics in the United States. Based upon his reputation in the history of mathematics he was appointed in 1918 to the first history of mathematics chair in the U. S, created for him, at the University of California, Berkeley, he remained in Berkeley, California until his death in 1930. Cajori did no original mathematical research unrelated to the history of mathematics. In addition to his numerous books, he contributed recognized and popular historical articles to the American Mathematical Monthly, his last work was a revision of Andrew Motte's 1729 translation of Newton's Principia, vol.1 The Motion of Bodies, but he died before it was completed. The work was finished by R. T. Crawford of Berkeley, California. 1917–1918, Mathematical Association of America president 1923, American Association for the Advancement of Science vice-president 1924, Invited Speaker of the International Congress of Mathematicians in 1924 in Toronto 1924–1925, History of Science Society vice-president 1929–1930, Comité International d'Histoire des Sciences vice-president The Cajori crater on the Moon was named in his honour 1890: The Teaching and History of Mathematics in the United States U.
S. Government Printing Office. 1893: A History of Mathematics, Macmillan & Company. 1898: A History of Elementary Mathematics, Macmillan. 1909: A History of the Logarithmic Slide Rule and Allied Instruments The Engineering News Publishing Company. 1916: William Oughtred: a Great Seventeenth-century Teacher of Mathematics The Open Court Publishing Company 1917: A History of Physics in its Elementary Branches: Including the Evolution of Physical Laboratories, The Macmillan Company. 1919: A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse, Open Court Publishing Company. 1920: On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century Vol. 1, University of California Press. 1928: A History of Mathematical Notations The Open Court Company. 1934: Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World tr. Andrew Motte, rev. Florian Cajori. Berkeley: University of California Press. 1913: "History of the Exponential and Logarithmic Concepts", American Mathematical Monthly 20: Page 5 From Napier to Leibniz and John Bernoulli I, 1614 — 1712 Page 35 The Modern Exponential Notation Page 75: The Creation of a Theory of Logarithms of Complex Numbers by Euler, 1747 — 1749 Page 107: From Euler to Wessel and Argand, 1749 — 1800, Barren discussion.
Page 148: Generalizations and refinements effected during the nineteenth century: Graphic representation Page 173: Generalizations and refinements effected during the nineteenth century Page 205: Generalizations and refinements effected during the nineteenth century These seven installments of the article are available through the Early Content program of Jstor. 1923: "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser. Vol. 25, No. 1, pp. 1–46 Works by Florian Cajori at Project Gutenberg Works by Florian Cajori at Faded Page Works by or about Florian Cajori at Internet Archive O'Connor, John J.. Florian Cajori at the Mathematics Genealogy Project Florian Cajori. A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse. BiblioBazaar. ISBN 978-1-143-05698-7
Joseph-Louis Lagrange
Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences, he remained in France until the end of his life. He was involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations, he proved. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series, he studied the three-body problem for the Earth and Moon and the movement of Jupiter's satellites, in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, presented the so-called mechanical "principles" as simple results of the variational calculus.
Born as Giuseppe Lodovico Lagrangia, Lagrange was of French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, married an Italian, his mother was from the countryside of Turin. He was raised as a Roman Catholic, his father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, Lagrange seems to have accepted this willingly, he studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident.
Alone and unaided he threw himself into mathematical studies. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange proved to be a problematic professor with his oblivious teaching style, abstract reasoning, impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex. Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Lagrange wrote several letters to Leonhard Euler between 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and simplifying Euler's earlier analysis. Lagrange applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was impressed with Lagrange's results, it has been stated that "with characteristic courtesy he withheld a paper he had written, which covered some of the same ground, in order that the young Italian might have time to complete his work, claim the undisputed invention of the new calculus". Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. In 1758, with the aid of his pupils, Lagrange established a society, subsequently incorporated as the Turin Aca
Leonhard Euler
Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.
He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.
In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.
In 1755