The post Lejeune Dirichlet and the Mathematical Function appeared first on SciHi Blog.

]]>On February 13, 1805, German mathematician **Johann Peter Gustav Lejeune Dirichlet** was born. Dirichlet is best known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. He also proposed in 1837 the modern definition of a mathematical function.

“In mathematics as in other fields, to find one self lost in wonder at some manifestation is frequently the half of a new discovery.”

– Gustav Lejeune Dirichlet, [13]

Gustav Lejeune Dirichlet was born in Düren, halfway between Aachen and Cologne in Germany, a town on the left bank of the Rhine which at the time was part of Napoleon Bonaparte’s French Empire. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor, whose family came from Richelet, a small town in Belgium. This explains the origin of his name which comes from “*Le jeune de Richelet*” meaning “*Young from Richelet*“. The youngest of seven children, his parents enrolled him in an elementary school and then private school to become a merchant. But, young Dirichlet showed a strong interest in mathematics and convinced his parents to allow him to continue his studies. In 1817 he was sent to the Gymnasium in Bonn and in 1820, Dirichlet moved to the Jesuit Gymnasium in Cologne, where his lessons with Georg Simon Ohm widened his knowledge in mathematics.[5]

By the age of 16 Dirichlet had completed his school qualifications and was ready to enter university. However, the standards in German universities were not high at this time so Dirichlet decided to study in Paris.[1] In 1822, Dirichlet contracted smallpox, but it did not keep him away from his lectures in the Collège de France for long. In 1823 he was recommended to General Foy, who hired him as a private tutor to teach his children German, the wage finally allowing Dirichlet to become independent from his parents’ financial support. Dirichlet’s first original research brought him immediate fame, since it was on Fermat’s Last Theorem.[6] The theorem claimed that for n > 2 there are no non-zero integers x, y, z such that x^{n} + y^{n} = z^{n}. The cases n = 3 and n = 4 had been proven by Leonard Euler and Fermat, and Dirichlet attacked the theorem for n = 5.[7] Adrien-Marie Legendre, one of the referees, soon completed the proof for this case, while Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case n=14.[8]

In 1825, encouraged by Alexander von Humboldt, Dirichlet decided to return to Germany. In order to teach in a German university Dirichlet would have needed an habilitation. Although Dirichlet could easily submit an habilitation thesis, this was not allowed since he did not hold a doctorate, nor could he speak Latin, as it was required by that time. The problem was nicely solved by the University of Cologne giving Dirichlet an honorary doctorate, thus allowing him to submit his habilitation thesis to the University of Breslau [1].

With Alexander von Humboldt’s help Dirichlet then moved to Berlin in 1828, where he was appointed at the Military College and would be able to teach at the University of Berlin. Soon after this he was appointed a professor at the University of Berlin where he remained from 1828 to 1855. In 1829, during a trip, he met Carl Gustav Jacob Jacobi, at the time professor of mathematics at Königsberg University.[9] Over the years they kept meeting and corresponding on research matters, in time becoming close friends. Dirichlet was appointed to the Berlin Academy in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn’s two sisters. In 1837 Dirichlet proposed the modern concept of a function y = f (x) in which for every x, there is associated with it a unique y. This work was inspired by significant contributions he had made to the understanding on Fourier Series, particularly with respect to conditions of convergence.[4] In mechanics he investigated the equilibrium of systems and potential theory, which led him to the Dirichlet problem concerning harmonic functions with prescribed boundary values.[3]

Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Leopold Kronecker that he had thirteen lectures a week to give in addition to many other duties.[10] It was therefore something of a relief when, on Carl Friedrich Gauss’s death in 1855, he was offered his chair at Göttingen.[1] Ernst Eduard Kummer was called to assume his position as a professor of mathematics in Berlin. The quieter life in Göttingen seemed to suit Dirichlet. He had more time for research and some outstanding research students. However, sadly he was not to enjoy the new life for long. Dirichlet died on 5 May 1859, in Göttingen. His lectures on number theory were edited by Richard Dedekind after his death and were given a famous appendix of their own. Dirichlet was known in his day for the (by the standards of the time) rigour of his proofs. Carl Gustav Jacobi wrote in a letter to Alexander von Humboldt on 21 December 1846:

“If Gauss says he has proved something, it is very probable to me that if Cauchy says it, it is just as much pro as contra to bet; if Dirichlet says it, it is certain.”

At yovisto academic video search, you can learn more about number theoryin a lecture on Gauss’ Law.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Peter Gustav Lejeune Dirichlet“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Elstrodt, Jürgen (2007). “The Life and Work of Gustav Lejeune Dirichlet (1805–1859)” (PDF). Clay Mathematics Proceedings
- [3] Peter Gustav Lejeune Dirichlet at Encyclopedia Britannica
- [4] Peter Taylor, Peter Gustav Lejeune Dirichlet at Australian Mathematics Trust (2001)
- [5] Georg Simon Ohm – The Misjudged Math Teacher, SciHi Blog, July 6, 2012.
- [6] Pierre de Fermat and his Last Problem, SciHi Blog, January 12, 2018.
- [7] Read Euler, he is the Master of us all…, SciHi Blog, September 18, 2015.
- [8] Legendre’s Elements of Geometry, SciHi Blog, September 18, 2014.
- [9] Carl Jacobi and the Elliptic Functions, SciHi Blog, December 10, 2014.
- [10] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi Blog, December 7, 2014
- [11] Lejeune Dirichlet at zbMATH
- [12] Lejeune Dirchlet at Wikidata
- [13] Peter Gustav Lejeune Dirichlet
*, Werke*, Bd. 8 (1897), 233. - [14] Timeline of Number Theorists, via DBpedia and Wikidata

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]]>The post Euclid of Alexandria – the Father of Geometry appeared first on SciHi Blog.

]]>At about 330 BC, **Euclid of Alexandria** was born, who often is referred to as the Father of Geometry. His *Elements* is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the *Elements*, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms.

“Ὅπερ ἔδει δεῖξαι.” (Which was to be proved. Latin translation: Quod erat demonstrandum (often abbreviated Q.E.D.).

— Euclid, Elements, Book I, Proposition 4.

Very few original references to Euclid survive, so little is known about his life. The date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other figures mentioned alongside him. The few historical references to Euclid were written long after he lived, by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD. Proclus introduces Euclid only briefly in his *Commentary* on the *Elements*. According to Proclus, Euclid belonged to Plato‘s “persuasion” and brought together the *Elements*, drawing on prior work by several pupils of Plato. Proclus believes that Euclid must have lived during the time of Ptolemy I because he was mentioned by Archimedes, who refers to him as the author of the *Elements.*[4]

Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid’s *Elements*, Euclid replied “*there is no royal road to geometry.*” This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great. Arabian authors state that Euclid was the son of Naucrates and that he was born in Tyre. It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors.

Because the lack of biographical information is unusual for the period, some researchers have proposed that Euclid was not, in fact, a historical character and that his works were written by a team of mathematicians who took the name Euclid from a historical figure. They think, Euclid might be similar to Bourbaki, the collective pseudonym under which a group of 20th-century mathematicians published with the aim of reformulating mathematics. However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.

*Euclid’s Elements*

Euclid‘s most famous work is his treatise on mathematics *The Elements*. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him.[1] One of Euclid‘s accomplishments was to present the material in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later. There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are “from the edition of Theon” or the “lectures of Theon“. The only reference that historians rely on of Euclid having written the Elements actually was from Proclus, who briefly in his *Commentary on the Elements* ascribes Euclid as its author.

In his *Elements*, Euclic put the mathematical knowledge of his age on a solid foundation. He began in Book I with 23 definitions, such as “*a point is that which has no par*t” and “*a line is a length without breadth*”, followed by five unproved assumptions that he called postulates (now known as axioms).[3] Euclid stated that axioms were statements that were just believed to be true, but he realized that by blindly following statements, there would be no point in devising mathematical theories and formulae. He realized that even axioms had to be backed with solid proofs.[2]

The *Elements* is divided into 13 books. Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Book six looks at applications of the results of book five to plane geometry.

Although best known for its geometric results, the *Elements* also includes number theory, which he presented in book seven to nine. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid‘s lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Books eleven to thirteen deal with three-dimensional geometry, in Greek stereometria. The immense impact of the Elements on Islamic mathematics is visible through the many translations into Arabic from the 9th century forward. Euclid first became known in Europe through Latin translations of these versions. The first extant Latin translation of the Elements was made about 1120 by Adelard of Bath, who obtained a copy of an Arabic version in Spain, where he traveled while disguised as a Muslim student. More than one thousand editions of The *Elements* have been published since it was first printed in 1482. The impact of this activity on European mathematics cannot be overestimated. The ideas and methods of Johannes Kepler,[5] Pierre de Fermat,[6] René Descartes, [7] and Isaac Newton were deeply rooted in, and inconceivable without, Euclid’s *Elements*.[1,3]

At yovisto academic video search, you can learn more about Non-Euclidian geometry in the History of Mathematics lecture of Professor N. J. Wildberger “*MathHist12 – Non-Euclidian Geometry*“.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Euclid of Alexandria“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Biography of Euclid at TheFamousPeople
- [3] Euclid at Encyclopedia Britannica
- [4] Archimedes lifted the world off their hinges, SciHi Blog
- [5] And Kepler Has His Own Opera – Kepler’s 3rd Planetary Law, SciHi Blog
- [6] Pierre de Fermat and his Last Problem, SciHi Blog
- [7] Cogito Ergo Sum – The Philosophy of René Descartes, SciHi Blog
- [8] Works by or about Euclid at Internet Archive
- [9] Works about or by Euclid, via Wikisource
- [10] Euclid at Wikidata
- [11] Timeline of mathematicians of the Hellenistic Epoch, via Wikidata

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]]>The post Ernst Kummer and the Introduction of Ideal Numbers appeared first on SciHi Blog.

]]>On January 29, 1810, German mathematician **Ernst Eduard Kummer** was born. One of his major contributions is the introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic to complex number fields.He also discovered the fourth order surface based on the singular surface of the quadratic line complex. This Kummer surface has 16 isolated conical double points and 16 singular tangent planes.

“A peculiar beauty reigns in the realm of mathematics, a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind, which has acquired an appreciation of it, very much like the latter. “

— Ernst Eduard Kummer [10]

Ernst Kummer was born in Sorau, Brandenburg (then part of Prussia)to his father Carl Gotthelf Kummer, a physician. However his father died when Ernst was only three years old and Ernst and his elder brother Karl were brought up by their mother. Ernst received private coaching before entering the Gymnasium in Sorau when he was nine years old. In 1828 Kummer entered the University of Halle to study Protestant theology. But fortunately for the good of mathematics, Kummer received mathematics teaching as part of his degree to provide a proper foundation to the study of philosophy. Soon the study of mathematics should become his main subject, although at this stage he still saw it as leading to a later study of philosophy.[1]

In 1831 Kummer was awarded a prize for a mathematical essay and in the same year he was awarded his certificate to enable him to teach in schools. Moreover, on the strength of his prize winning essay, he was awarded a doctorate. After a probationary year at the Gymnasium in Sorau where he had been educated, he was appointed to a teaching post at the Gymnasium in Liegnitz, now Legnica in Poland that he held for the next 10 years. Some of his pupils there had great ability, most prominent among them Leopold Kronecker, and conversely they were extremely fortunate to find a school teacher of Kummer‘s quality and ability to inspire. During Kummer‘s first period of mathematics he worked on function theory. He extended Gauss‘s work on hypergeometric series, giving developments that are useful in the theory of differential equations.[7] He was the first to compute the monodromy groups of these series. In 1839, although still a school teacher, Kummer was elected to the Berlin Academy on Dirichlet‘s recommendation.

In 1842, Kummer became professor of mathematics at the University of Breslau (now Wrocław, Poland). In 1855 he succeeded Peter Gustav Lejeune Dirichlet [8] as professor of mathematics at the University of Berlin, at the same time also becoming a professor at the Berlin War College. In 1843, Kummer showed Dirichlet an attempted proof of Fermat’s last theorem. But, Dirichlet found an error, and Kummer continued his search and developed the concept of so-called ideal numbers. In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field. An ideal is a special subset of an algebraic ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number.

Using the concept of ideal numbers, Ernst Kummer proved the insolubility of the Fermat relation for all but a small group of primes, and he thus laid the foundation for an eventual complete proof of Fermat’s last theorem. For his great advance, the French Academy of Sciences awarded him its Grand Prize in 1857. The ideal numbers have made possible new developments in the arithmetic of algebraic numbers. In 1861, Germany’s first seminar in pure mathematics was established at Berlin on the recommendation of Kummer and Karl Weierstrass.[9] It soon attracted gifted young mathematicians form throughout the world, including many graduate students.[3] Kummer’s Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory.

For the rest of his life, Ernst Kummer was devoted to geometry. He applied himself with unbroken productivity to ray systems and also considered ballistic problems. By that time, also discovered the fourth order surface, now named after him, based on the singular surface of the quadratic line complex. The Kummer surface has 16 isolated conical double points and 16 singular tangent planes and was published in 1864. He retired at his own request in 1883 to spent the last years of his life in quiet retirement until his death in 1893 at age 83.

At yovisto academic video search, you can learn more about algebraic geometry in the lecture of Gregory Sankaran on *Moduli of deformation generalised Kummer varieties*.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Ernst Kummer“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Ernst Eduard Kummer at Encyclopedia Britannica
- [3] Kummer, Ernst Eduard, at Encyclopedia.com, Complete Dictionary of Scientific Biography
- [4] Ernst Eduard Kummer at zbMATH
- [5] Ernst Eduard Kummer at Mathematics Genealogy Project
- [6] Ernst Eduard Kummer at Wikidata
- [7] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog
- [8] Lejeune Dirichlet and the Mathematical Function, SciHi Blog
- [9] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog
- [10] Berliner Monatsberichte (1867), 395. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 185
- [11] Timeline for Ernst Eduard Kummer via Wikidata

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]]>The post Alfred Tarski and the Undefinability of Truth appeared first on SciHi Blog.

]]>On January 14, 1902, Polish-American mathematician and logician **Alfred Tarski** was born. A prolific author he is best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy. For my Semantic Web Technologies lecture series I always introduce my students to model-theoretic semantics as means to enable a formal representation of meaning for languages. I guess, they don’t like the mathematical overhead. But nevertheless, you need it to make sense of any logical expression.

“Logic is justly considered the basis of all other sciences, even if only for the reason that in every argument we employ concepts taken from the field of logic, and that ever correct inference proceeds in accordance with its laws.”

— Alfred Tarski, Introduction to Logic: and to the Methodology of Deductive Sciences. (1941/2013)

Born as Alfred Teitelbaum into a family of Polish Jews of comfortable circumstances, Tarski first manifested his mathematical abilities while in secondary school, at Warsaw’s Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology. After Poland regained independence in 1918, Warsaw University quickly became a world-leading research institution in logic, foundational mathematics, and the philosophy of mathematics. Famous Mathematician Stanisław Leśniewski recognized Tarski’s potential as a mathematician and encouraged him to abandon biology. Henceforth Tarski attended courses taught by Jan Łukasiewicz, Wacław Sierpiński, and became the only person ever to complete a doctorate under Leśniewski’s supervision.

In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski and also converted to Roman Catholicism, Poland’s dominant religion, even though Tarski was an avowed atheist. After completing his doctorate at Warsaw University 1923, Tarski served as Łukasiewicz’s assistant. Tarski’s first major results were published in 1924 when he began building on the set theory results obtained by Cantor, Zermelo and Dedekind. He published a joint paper with Banach in that year on what is now called the Banach-Tarski paradox.[1] Between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but also did so while supporting himself primarily by teaching high-school mathematics at Warsaw secondary school, because of the small salary at Warsaw University. In 1930, Tarski visited the University of Vienna, lectured to Karl Menger’s colloquium, and met Kurt Gödel.[4] Due to an invitation from Harvard University, Tarski was able to leave Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II.

Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his extended family died at the hands of the German occupying authorities. Thanks to a Guggenheim Fellowship, Tarski visited the Institute for Advanced Study in Princeton in 1942, where he again met Gödel, who also had fled from Nazi Germany. Subsequently, he joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career until he became emeritus in 1968.

Tarski was a charismatic teacher who charmed his students, but he demanded perfection and could be devastatingly abusive to those who failed to measure up.[3] He is recognised as one of the four greatest logicians of all time, the other three being Aristotle, Frege, and Gödel. Of these Tarski was the most prolific as a logician and his collected works, excluding his books, runs to 2500 pages. Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics. He produced axioms for ‘logical consequence‘, worked on deductive systems, the algebra of logic and the theory of definability. He can be considered a mathematical logician with exceptionally broad mathematical interests.[1]

One example of his achievements is a decision procedure for sentences written in the language of the arithmetic of real numbers. These are sentences that can be written using variables ranging over the real numbers, using symbols for the operations of addition and multiplication and for the relations of equality and order .[3]

“It is perhaps worth while saying that semantics as conceived in this paper (and in former papers of the author) is a sober and modest discipline which has no pretensions to being a universal patent-medicine for all the ills and diseases of mankind, whether imaginary or real. You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflicts. Nor is semantics a device for establishing that everyone except the speaker and his friends is speaking nonsense.”

— Alfred Tarski, The Semantic Conception of Truth (1952)

Another great achievement was his assault on the notion of truth. Tarski was able, under suitable conditions, to give a mathematically precise definition of what it means to say that a given sentence of a language is true. One of these conditions was that the syntax of the language in question be formally well-defined, i.e. one could say precisely just which expressions are legitimate sentences and which not. Moreover, a sentence had to have a well-defined semantics, i.e. the meaning of the individual components of the sentence had to be (formally) given. Now, the “metalanguage” in which this truth definition is developed is, in general, separate from the language whose true sentences are being identified. As Kurt Gödel previously had shown, it is possible for a language to function as its own metalanguage. But for this case, Tarski was able to prove his famous “undefinability theorem“: Under very general conditions, the notion of “truth” of the sentences of a language cannot be defined in that same language.[3] Thus, Tarski radically transformed Hilbert’s proof-theoretic metamathematics. He destroyed the borderline between metamathematics and mathematics by his objection to restricting the role of metamathematics to the foundations of mathematics

Alfred Tarski died on October 26, 1983, in Berkeley, California, at age 82.

At yovisto academic video search, you can learn more about the history of mathematical logics in the lecture of Prof Christos H. Papadimitriou on the Graphic Novel ‘*Logicomix: An Epic Search for Truth*‘.

**References and Further Reading**

- [1] Alfred Tarski at McTutor’s history of Mathematics
- [2] Alfred Tarski at biography.yourdictionary.com
- [3] Martin Davis: The Man who defined Truth, book review at americanscientist.org, 2005.
- [4] Kurt Gödel Shaking the Very the Foundations of Mathematics, SciHi Blog, April 28, 2012.
- [5] Alfred Tarski at zbMATH
- [6] Alfred Tarski at Mathematics Genealogy Project
- [7] Alfred Tarski at Wikidata
- [8] Tarski’s Truth Definitions by Wilfred Hodges, Stanford Encyclopaedia of Philosophy
- [9] Timeline of logicians, via Wikidata

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]]>The post Agner Erlang and the Mathematics of Telecommunication Traffic appeared first on SciHi Blog.

]]>On January 1, 1878, Danish mathematician, statistician and engineer **Agner Krarup Erlang **was born, who invented the fields of traffic engineering and queueing theory. He developed mathematical theories applying the theory of probability, while working for the Copenhagen Telephone Company. He provided significant insights for planning the operation of automatic telephone exchanges that proved so useful that his formulas were used by telephone companies in other countries.

Agner Erlang was born at Lønborg, near Tarm, in Jutland to Hans Nielsen Erlang, a schoolmaster and parish clerk, and Magdalene Krarup, who came from an ecclesiastical family but was descended from the mathematician Thomas Fincke. Agner was a bright child, learning quickly and having an excellent memory.[2] After his primary education, he was tutored at home by his father and another teacher from his father’s school. At age 14, he passed the Preliminary Examination of the University of Copenhagen with distinction, after receiving dispensation to take it because he was younger than the usual minimum age. For the next two years he taught alongside his father.

A distant relative provided free board and lodging, and Erlang prepared for and took the University of Copenhagen entrance examination in 1896, and passed with distinction. He won a scholarship to the University and majored in mathematics, and also studied astronomy, physics and chemistry. He graduated in 1901 with an MA and over the next 7 years taught at several schools. During this time he kept up his interest in mathematics, and he received an award in 1904 for an essay on Huygens‘ solution of infinitesimal problems which he submitted to the University of Copenhagen.[7]

Erlang’s interests turned towards the theory of probability and he kept up his mathematical interests by joining the Danish Mathematical Association. At meetings of the Mathematical Association he met Johan Ludwig Jensen who was then chief engineer at the Copenhagen Telephone Company, which Erlang joined in 1908 as a scientific collaborator. There, Erlang was presented with the classic problem of determining how many circuits were needed to provide an acceptable telephone service. His thinking went further by finding how many telephone operators were needed to handle a given volume of calls. Most telephone exchanges then used human operators and cord boards to switch telephone calls by means of jack plugs.

In 1909, Agner Erlang published “*The Theory of Probabilities and Telephone Conversations*” proving that telephone calls distributed at random follow Poisson’s law of distribution. Out of necessity, Erlang was a hands-on researcher. At the beginning he had no laboratory staff to help him, so he had to carry out all the measurements of stray currents. He was often to be seen in the streets of Copenhagen, accompanied by a workman carrying a ladder, which was used to climb down into manholes.[3] Further publications followed, the most important work was published in 1917 “*Solution of some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges*“, which contained formulae for loss and waiting time, which are now well known in the theory of telephone traffic.For example, the Erlang-B formula can be used to calculate the number of telephone lines needed in a call center – the Erlang-C formula can be used to estimate the number of call center agents needed for a given call volume. His papers were prepared in a very brief style and can be difficult to understand without a background in the field. One researcher from Bell Telephone Laboratories is said to have learned Danish to study them.

Agner Erlang’s work on the theory of telephone traffic won him international recognition. The British Post Office accepted his formula as the basis for calculating circuit facilities. Moreover, a unit of measurement, statistical distribution and programming language mainly used for large industrial real-time systems have been named in his honor. His name is also given to the statistical probability distribution that he used in his work. Erlang was also an expert in the history and calculation of the numerical tables of mathematical functions, particularly logarithms. Erlang worked for the Copenhagen Telephone Company for almost 20 years, and never having had time off for illness, went into hospital for an abdominal operation in January 1929.

In 1946, at the suggestion of David George Kendall, the International Consultative Committee on Telephones and Telegraphs (CCITT) (the predecessor of the ITU) named the basic unit for traffic in a communications network “Erlang” in his honour. A programming language developed by Joe Armstrong and others at the Swedish company Ericsson was also named after Erlang. However, Erlang can also be interpreted as “**Er**icsson **lang**uage”.

Agner Erlang died some days later on February 3, 1929.

At yovisto academic video search, you may be interested in a short conceptional introduction into the programming language Erlang, presented at the 24th conference of the Chaos Computer Club by Stefan Strigler.

**References and Further Reading: **

- [1] O’Connor, John J.; Robertson, Edmund F., “Agner Krarup Erlang“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Agner Krarup Erlang – plus magazine – living mathematics
- [3] Who was Agner Krarup Erlang? at informs.org
- [4] Agner Krarup Erlang at Wikidata
- [5] Agner Krarup Erlang at zbMATH
- [6] Agner Krarup Erlang at Wikidata
- [7] Christiaan Huygens and the Discovery of Saturn Moon Titan, SciHi Blog
- [8] “The Theory of Probabilities and Telephone Conversations”.
*Nyt Tidsskrift for Matematik*.**20**(B): 33–39. 1909. - [9] Timeline of Danish Engineers, via DBpedia and Wikidata

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]]>The post A great man whose only fault was being a woman – Émilie du Châtelet appeared first on SciHi Blog.

]]>On December 17, 1706, French mathematician, physicist, and author** Gabrielle Émilie Le Tonnelier de Breteuil, marquise du Châtelet** was born. Her major achievement is considered to be her translation and commentary on Isaac Newton‘s work* Principia Mathematica*, which still is the standard French translation of Newton‘s work today. Philosopher and author Voltaire, one of her lovers, once declared in a letter to his friend King Frederick II of Prussia that du Châtelet was “*a great man whose only fault was being a woman*“.

“To be happy, one must be free from prejudice; be virtuous; be well; have tastes and passions; be susceptible to illusion; for we owe most of our pleasures to illusion, and unfortunate is the one who loses it.”

– Emilie du Châtelet, Opuscules philosophiques et littéraires, 1769

Émilie du Châtelet was born in Paris, the only daughter of six children to Louis Nicolas le Tonnelier de Breteuil, a member of the lesser nobility. At the time of du Châtelet’s birth, her father held the position of the Principal Secretary and Introducer of Ambassadors to King Louis XIV. He held a weekly salon on Thursdays, to which well-respected writers and scientists were invited. Du Châtelet’s education has been the subject of much speculation, but nothing is known with certainty. Among the acquaintances of Nicolas le Tonnelier de Breteuil was Fontenelle, the perpetual secretary of the French Académie des Sciences. Recognizing Émilie’s early brilliance and abilities, he arranged for Fontenelle to visit and talk about astronomy with her when she was 10 years old. Her father arranged training for her in physical activities such as fencing and riding, and as she grew older, he brought tutors to the house for her to learn Latin, Italian, Greek and German as well as mathematics, literature, and science.

In 1725, nineteen year old Émilie married the 15 year older Marquis Florent-Claude du Chastellet-Lomont, conferring her the title of Marquise du Chastellet. Like many marriages among the nobility, the marriage was arranged. After the birth of her third child in 1733, Émilie resumed her mathematical studies in algebra and calculus with Moreau de Maupertuis, a member of the Academy of Sciences and student of Johann Bernoulli, one of the many prominent mathematicians in the Bernoulli family,[7] followed by prominent French mathematician, astronomer, geophysicist, and intellectual Alexis Clairaut, a mathematical prodigy. In the very same year, Émilie started a friendship and love affair with Voltaire,[8] whom she had met first already in 1729. Émilie invited Voltaire to live in her country house at Cirey-sur-Blaise, northeastern France, and he became her long-time companion — under the eyes of her tolerant husband. There she studied physics and mathematics and published scientific articles and translations.

In the frontispiece to her translation of Newton,[9] du Châtelet is depicted as the muse of Voltaire, reflecting Newton’s heavenly insights down to Voltaire. A major contribution to physics was Émilie du Châtelet’s advocacy of kinetic energy. Although in the early 18th century the concepts of force and momentum had been long understood, the idea of energy as a transferrable currency between different systems, was still in its infancy. Inspired by theoretical work of Gottfried Wilhelm Leibniz,[10] Émilie du Châtelet repeated and publicized an experiment originally devised by Willem ‘s Gravesande in which balls were dropped from different heights into a sheet of soft clay. Each ball’s kinetic energy – as indicated by the quantity of material displaced – was shown conclusively to be proportional to the square of the velocity. Earlier workers like Newton and Voltaire had all believed that “energy” was indistinct from momentum and therefore proportional to velocity. But, what Émilie du Châtelet is best remembered today, is her translation of Newton’s *Principia Mathematica* into French, including her commentaries and her derivation of the notion of conservation of energy from its principles of mechanics. Today, du Châtelet’s translation of *Principia Mathematica* is still the standard translation of the work into French.

From 1744 to 1748 she spent part of her time in Versailles together with Voltaire, who thanks to Madame de Pompadour had regained access to the farm. In the years 1748/49 she often lived with him in Lunéville Castle at the court of Stanislaus I. Leszczyński, the father-in-law of Louis XV and Polish ex-king, who had been compensated in 1735 by the Duchy of Lorraine. Here she began an affair with the courtier, officer and poet Jean François de Saint-Lambert. When she became pregnant, she succeeded, together with Saint-Lambert and Voltaire (who in turn had been in contact with a widowed niece since 1745), in convincing her husband that the child was his. On the night of 3 September 1749, she gave birth to a daughter, Stanislas-Adélaïde, but died a week later, at Lunéville, from a pulmonary embolism.

At yovisto academic video search, you can learn more about the life and work of Émilie du Châtelet in a German language lecture of Prof Ruth Hagengruber.

**References and Further Reading: **

- [1] David Bodanis: Passionate Minds: The Great Enlightenment Love Affair, — a novel dealing with the life and love of Voltaire and his mistress, scientist Émilie du Châtelet.
- [2] David Bodanis: The Scientist whom History forgot, The Guardian, Aug 4, 2006.
- [3] Émilie du Châtelet at Biographies of Women Mathematicians
- [4] A Love Story — Voltaire and Émilie
- [5] Émilie du Châtelet at Stanford Encyclopedia of Philosophy
- [6] Émilie du Châtelet at Wikidata
- [7] How to calculate fortune – Jakob Bernoulli, SciHi Blog
- [8] Voltaire – Libertarian and Philosopher, SciHi Blog
- [9] Standing on the Shoulders of Giants – Sir Isaac Newton, SciHi Blog
- [10] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [11] Timeline of women in mathematics, via Wikidata

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]]>The post Hypatia – the first Woman in Mathematics appeared first on SciHi Blog.

]]>The Neoplatonian philosopher **Hypatia of Alexandria**, Egypt, was the first well-documented woman in mathematics. Her actual date of birth is unknown, although considered somewhen between 350 and 370 AD. She was the head of the Platonist school at Alexandria and additionally taught philosophy and astronomy.

There is little news about Hypatia’s life and work. Hypatia’s father was the astronomer and mathematician Theon of Alexandria, the last scientist known by name in the Museion of Alexandria, a famous government-funded research centre. Hypatia was probably born around 355, for at the time of her death she was, as the chronicler Johannes Malalas reports, already an “old woman”, presumably about sixty years old.[3] She seems to have spent her whole life in her home town of Alexandria. She received mathematical and astronomical training from her father. Later she participated in his astronomical work. Who her philosophy teacher was is unknown.

After completing her education, she began to teach mathematics and philosophy herself. According to the Suda, she combined rhetorical talent with a prudent, thoughtful approach. Socrates of Constantinople reports that listeners have come to her from everywhere. Some of her students were Christians. The most famous of them was Synesius, who studied philosophy and astronomy with her in the last decade of the 4th century. Damaskios reports that Hypatia wore the philosopher’s mantle (tríbōn) and travelled the city to teach publicly and interpret the teachings of Plato or Aristotle or any other philosopher to anyone who wanted to hear them.

It is known that people from far away came just to be taught by her and she hosted numerous public lectures, open for everyone who wanted to hear about her research in philosophy, mathematics and astronomy. Unfortunately, many contemporary scientists disliked this behavior, thinking that philosophy should not be taught to anyone who was not adequately educated and that teachers should not just ‘walk around’ town and teach anything openly, especially not women. Hypatia also hosted events with only very small groups of people, who were prohibited to talk about anything that was discussed during these meetings.

An anecdote handed down in the Suda also points in this direction, according to which she showed a pupil in love with her her menstrual blood as a symbol of the impurity of the material world, in order to drastically show him the questionability of his sexual desire. The disregard for the body and physical needs was a feature of the neoplatonic world view. Although the anecdote may have originated in legend, it may have a true core; in any case, Hypatia was known not to shy away from consciously provocative behavior. This is also an indication of a cynical element in their philosophical stance: cynics used to shock in a calculated way in order to bring about knowledge.

In addition to the subject matter that Hypatia taught to the public, there were also secret teachings that were to be reserved for a smaller circle of students. This is evident from the correspondence of Synesius, who repeatedly reminds his friend and classmate Herkulianos of the commandment of secrecy (echemythía) and accuses Herkulianos of not having kept it. Synesios refers to the Pythagorean imperative of silence; the transmission of secret knowledge to unqualified persons leads to such vain and incomprehensible listeners in turn passing on what they have heard in a distorted form, which ultimately leads to a discrediting of philosophy in public.

Even though it is not clear, how old Hypatia exactly was when she passed away, but her life sure ended quickly. She was murdered in spring of 415 or 416, which also is not clear. The assumed motives for the crime were personal, religious and political. During the second half of the 4th century, religious fights between Christians and followers of other rituals got more frequent. The fights escalated and Cyril of Alexandria is assumed to have spread false rumors about Hypatia whom he disliked due to her religious and political believes. However, many people of Alexandria were spurred on to take revenge on the woman. She was carried into a church, brutally killed and finally burnt.

During her life time, Hypatia published next to her teaching activities several works on mathematics and astronomy, especially in the field of arithmetics and conic sections. Unfortunately, not a single publication by Hypatia was found and therefore historians have no proof that she published them up to this day. Clear is however, that her father described Hypatia’s achievements in various works and that her brilliance was admired by many. The question of which direction of New Platonism Hypatia belonged to is answered differently in research. Since the sources do not provide anything, only hypothetical considerations are possible. According to one assumption, the philosopher placed herself in the tradition of Iamblicho and accordingly conducted theurgy. According to the contrary opinion she was rather part of the direction of Plotin and Porphyrios, which postulated a salvation of the soul by its own power through spiritual striving for knowledge.

But despite the many critics during her life time and today, Hypatia was respected for her scientific achievements by numerous scientists. Also in later years, authors and scientists discussed the case of Hypatia. Voltaire once even described her as one of the earliest distributors of the thoughts of the Enlightenment and saw her murder as a proof of the church’s fanaticism. Historians assume on this day that her sudden death caused a great gap concerning her fields of study and especially the role of women in science. Even Bertrand Russel‘s wife, Donna Russel, published a scientific work on Hypatia’s history and her achievements. Based on Hypatia’s life, many musical and literary works were published and performed throughout history, including novels, operas and poems.

At yovisto academic video search, you may be interested in a video lecture by Professor Maria Dzielska, who discusses Hypatia’s achievements at Poland’s Embassy in Washington D.C.

**References and Further Reading:**

- [1] Article about Hypatia at women-philosophers.com
- [2] Parsons, Reuben. St. Cyril of Alexandria and the Murder of Hypatia
- [3] Whitfield, Bryan J. The Beauty of Reasoning: A Reexamination of Hypatia and Alexandria
- [4] Socrates of Constantinople,
*Ecclesiastical History*, VII.15, at the Internet Archive - [5] Zielinski, Sarah (14 March 2010), “Hypatia, Alexandria’s Great Female Scholar”,
*Smithsonian* - [6] Works by or about Hypatia at Wikisource
- [7] Richeson, A. W. (1940), “Hypatia of Alexandria” (PDF),
*National Mathematics Magazine*,**15**(2): 74–82 - [8] Hypatia at Wikidata
- [9] Timeline of Ancient Greek Women Philosophers, via DBpedia and Wikidata

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]]>The post George Boole – The Founder of Modern Logics appeared first on SciHi Blog.

]]>On December 8, 1864, British mathematician and logician **George Boole** passed away. He is best known as the inventor of the prototype of what is now called Boolean logic, which became the basis of the modern digital computer. Thus, Boole also is regarded as one of the founders of the field of computer science.

“It appeared to me that, although Logic might be viewed with reference to the idea of quantity, it had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the medium of Number with the intuitions of Space and Time, it was lawful also to regard it from within, as based upon facts of another order which have their abode in the constitution of the Mind.”

— George Boole, The Mathematical Analysis of Logic, 1847

George Boole was born on November 2, 1815 as first of four children to his father John Boole, a London tradesman who was interested in science and in particular the application of mathematics to scientific instruments, and his wife Mary Ann Joyce, a lady’s maid. The family were not well off, partly because John’s love of science and mathematics meant that he did not devote the energy to developing his business in the way he might have done. George Boole had an elementary school education, but little further formal and academic teaching. William Brooke, a bookseller in Lincoln, has introduced him into the Latin language, when George went on to teach himself Greek.

By the age of 14 he had become so skilled in Greek that it provoked an argument. He translated a poem by the Greek poet Meleager which his father was so proud of that he had it published. However the talent was such that a local schoolmaster disputed that any 14 year old could have written with such depth. By that time he had entered the school of Thomas Bainbridge, the Bainbridge’s Commercial Academy in Lincoln. This school did not provide the type of education he would have wished but it was all his parents could afford. However he was able to teach himself French and German studying for himself academic subjects that a commercial school did not cover.

At age 16 Boole took up a junior teaching position in Doncaster, at Heigham’s School. This was rather forced on him since his father’s business collapsed and he found himself having to support financially his parents and his younger siblings. He maintained his interest in languages, began to study mathematics seriously. In 1833 he moved to a new teaching position in Liverpool but he only remained there for six months before moving to Hall’s Academy in Waddington, four miles from Lincoln. In 1834 he opened his own school in Lincoln although he was only 19 years old. Four years later he took over Hall’s Academy, at Waddington, outside Lincoln, following the death of Robert Hall. In 1840 he moved back to Lincoln, where he ran a boarding school. From 1838 onwards Boole was making contacts with sympathetic British academic mathematicians, and reading more widely. He studied algebra in the form of symbolic methods, as these were understood at the time, and began to publish research papers. Boole’s status as mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queen’s College, Cork in Ireland. He taught there for the rest of his life, gaining a reputation as an outstanding and dedicated teacher.

“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method;”

– George Boole, An Investigation of the Laws of Thought (1854)

Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles of Aristotle‘s logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability. Boole’s initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton [5] who supported the theory of “quantification of the predicate”, and Boole’s supporter Augustus De Morgan [6] who advanced a version of De Morgan duality, as it is now called. Boole’s approach was ultimately much further reaching than either sides’ in the controversy. It founded what was first known as the “algebra of logic” tradition. Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic.[7] In his 1847 publication *The Mathematical Analysis of Logic*, Boole created the first algebraic logic calculus and thus founded modern mathematical logic, which differs from the logic of the time by consistent formalisation. He formalised classical logic and propositional logic and developed a decision-making procedure for the true formulae over a disjunctive normal form.

In 1854 Boole published his most important work ‘*An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities*‘. Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics. He pointed out the analogy between algebraic symbols and those that represent logical forms. It began the algebra of logic called Boolean algebra which now finds application in computer construction, switching circuits etc.

“It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its object and in its instruments it must at present stand alone.”

– George Boole, The Mathematical Analysis of Logic, 1847

Many honours were given to Boole as the genius in his work was recognised. He received honorary degrees from the universities of Dublin and Oxford and was elected a Fellow of the Royal Society (1857). However his career, which was started rather late, came to an unfortunately early end when he died of fever-induced pleural effusion at the age of 49 in 1864. Boolean algebra has wide applications in telephone switching and the design of modern computers and his work has to be seen as a fundamental step in today’s computer revolution.

At yovisto academic video search you might learn more about Boolean Algebra in the lecture of Prof. Jim Pytel from Columbia Gorge Community College.

**References and further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson:
*George Boole.*In:*MacTutor History of Mathematics archive.* - [2] Des MacHale:
*George Boole: His Life and Work*. Boole Press (1985) - [3] Works by and about George Boole, via Wikisource
- [4] Works by or about George Boole at Internet Archive
- [5] William Hamilton and the Quaterions, SciHi Blog
- [6] Augustus de Morgan and Formal Logic, SciHi Blog
- [7] Gottlob Frege and the Begriffsschrift, SciHi Blog
- [8] George Boole: A 200-Year View by Stephen Wolfram November 2015
- [9] George Boole at Wikidata
- [10] Boole:
*An Investigation of The Laws of Thought*, London 1854, - [11] George Boole at zbMATH
- [12] Timeline of logicians, via Wikidata

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]]>The post August Ferdinand Möbius and the Beauty of Geometry appeared first on SciHi Blog.

]]>On November 17, 1790, German mathematician and astronomer **August Ferdinand Möbius** was born. He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space.

August Möbius’ father Johann Heinrich Möbius was a dance teacher in Schulpforta, near Naumburg on the Saale River in the German state of Saxony-Anhalt. He died three years after the birth of August Ferdinand. The mother Johanne Katharine Christiane Keil (1756-1820) was a descendant of Martin Luther.[5] After the early death of his father, his mother took care of the school education herself. August Möbius’ interest in mathematics evolved early, but it is said that his teacher hid several mathematical books from the curious boy to guarantee a widely ranged education instead of mathematics only. He attended the traditional regional school Pforta in his native town and graduated from high school there. He first studied law before turning to studying mathematics at the University of Leipzig in the second semester 1809 to 1814. He is particularly influenced by his astronomy professor Karl Mollweide, who is also the discoverer of a true to area map projection and some trigonometric formulas. In the meantime he attended lectures in Göttingen with Carl Friedrich Gauss and in Halle with Johann Friedrich Pfaff, one of Gauss’s teachers. He received his doctorate under Johann Friedrich Pfaff with the topic *De computandis occultationibus fixarum per planetas,* (*calculation methods for the occultation of fixed stars by planets)*.

Through a scholarship he was then able to work for Karl Friedrich Gauss,[4] who is known to have influenced the young scientist significantly. After his habilitation, which he earned with the work titled “*De peculiaribus quibusdam aequationum trigonometricarum*” in 1815, on the recommendation of Carl Friedrich Gauss, he was appointed extraordinary professor and observer of the Leipzig Observatory.

Having achieved all of this, Möbius was not even 26 years old, but further promotions were difficult. Möbius was known to be very unpretentious, extremely scientifically accurate and in love with Saxony, wherefore he refused several job offers all over Europe. Eventually, he accepted a position as a professor at the University of Jena, where he worked for the rest of his life. He was appointed director of the observatory in 1848. From 1846 he was a member of the Göttingen Academy of Sciences. From 1840 onwards he dealt with topological questions; the five-prince problem he posed has entered the literature: *once upon a time there was a king who had five sons. Since he could not choose one of his sons as his successor, he decided that after his death the kingdom should be divided into five parts in such a way that each part had a common border with each other*. Can this condition be fulfilled? Problems of this kind are not taken up by other mathematicians until a few years later, as e.g. the four colouring of maps as asked by Francis Guthrie. The answer, of course, is negative and easy to show.

It took him about 10 years to write his famous geometry book “*Der barycentrische Calcul – ein neues Hilfsmittel zur analytischen Behandlung der Geometrie*” (1827) and it was a great success in the scientific community in Germany and beyond. In it he developed the “*theory of geometric relationships of figures*” (equality, similarity, affinity, collineation), which was later used by Felix Klein to classify the various geometric approaches (Erlanger Program).[7] Further publications by August Möbius in the field of mathematics and astronomy were also quite influential and caused him a pretty good reputation during his lifetime. Möbius wrote numerous extensive treatises and writings on astronomy, geometry and statics. He made valuable contributions to analytic geometry, including the introduction of homogeneous coordinates and the principle of duality. He is regarded as a pioneer of topology. One of the most important mathematical objects, Möbius worked on was the famous “Möbius Strip”.

The “Möbius strip” in general is a surface with just one side and one boundary component. A Möbius strip is a two-dimensional surface with only one side. It can be constructed in three dimensions as follows. Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180 degree twist. It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A. Also, it has the mathematical property of being non-orientable. Johann Benedict Listing and August Möbius discovered this independently from each other in 1858, still, the object was named after Möbius. This work played and plays a significant role in physics, philosophy, chemistry and even certain double helix are structured as a Möbius strip. Möbius also contributed to the development of various areas of mathematics in numerous smaller articles. The Möbius geometry, the Möbius function, and the Möbius transformation are also named after him.

At yovisto academic video search, you may be interested in a video lecture titled “*Non-orientable surfaces – the Mobius band*” by Professor N J Wildberger of the School of Mathematics and Statistics at UNSW.

**References and Further Reading:**

- [1] August Möbius at the University of Munich [PDF]
- [2] O’Connor, John J.; Robertson, Edmund F., “August Ferdinand Möbius”,
*MacTutor History of Mathematics archive*, University of St Andrews. - [3] August Ferdinand Möbius at zbMATH
- [4] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog
- [5] Martin Luther – Iconic Figure of the Reformation, SciHi Blog
- [6] August Ferdinand Möbius at the Mathematics Genealogy Project
- [7] Felix Klein and the Klein-Bottle, SciHi Blog
- [8] Heinz Klaus Strick,
*August Ferdinand Möbius (1790–1868)*, Der Mathematische Monatskalender, Spektrum.de*(in German)* - [9] C. Bruhns:
*Todes-Anzeige.*(Obituary) Astronomische Nachrichten, Bd. 72 (1868), S. 287.*(in German)* - [10] August Ferdinand Möbius – Œuvres complètes Gallica-Math
- [11] August Ferdinand Möbius at Wikidata
- [12] Timeline for August Ferdinand Möbius, via Wikidata

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]]>The post Leibniz and the Integral Calculus appeared first on SciHi Blog.

]]>On November 11, 1675, German mathematician and polymath **Gottfried Wilhelm Leibniz** demonstrates integral calculus for the first time to find the area under the graph of y = ƒ(x). Integral calculus is part of infinitesimal calculus, which in addition also comprises differential calculus. In general, infinitesimal calculus is the part of mathematics concerned with finding tangent lines to curves, areas under curves, minima and maxima, and other geometric and analytic problems. Today, Gottfried Wilhelm Leibniz as well as independently Sir Isaac Newton are considered to be the founders of infinitesimal calculus. We already dedicated an article at the SciHi blog to Leibniz and his works.[9] But, Leibniz was kind of a universal polymath. His achievements are so numerous that we will definitely have more articles in the future about his contributions to science.

“Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum’s composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysic except he who has passed through this [labyrinth].”

– Wilhelm Gottfried Leibniz, Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae (Spring 1676)

To determine the area of curved objects or even the volume of a physical body with curved surfaces is a fundamental problem that has occupied generations of mathematicians since antiquity. For approximation, you don’t need modern integral calculus to solve this problem. Even the ancient Greeks had developed a method to determine integrals via the method of exhaustion, which also is the first documented systematic technique capable of computing areas and volumes. The method of exhaustion was described by the ancient Greek astronomer Eudoxus (ca. 370 BC), who tried to determine areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. But also in the far east, the Chinese independently developed similar methods around the 3rd century AD by Liu Hui, who used it to find the area of the circle. Later used in the 5th century Liu Hui‘s method was further developed by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.

Further improvement took until the rise of the European renaissance, when Italian mathematician Bonaventura Cavalieri in the 17th century developed his method of indivisibles.[10,1] In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. Such elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri‘s method. As an application, he computed the areas under the curves *y=x ^{n}* – up to the degree 9 – which is known as Cavalieri‘s quadrature formula. Together with the work of Pierre Fermat, they began to lay the foundations of modern calculus. Further steps were made by English theologian and mathematician Isaac Barrow and Italian physicist and mathematician Evangelista Torricelli,[11,4] who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus, which links the concept of the derivative of a function with the concept of the integral. English mathematician John Wallis generalized Cavalieri‘s method, computing integrals of x to a general power, including negative powers and fractional powers.

But, the major advance in integration came with the discoveries and development by Newton and Leibniz. It was Barrow‘s student Isaac Newton,[12] who completed the development of the fundamental theorem of calculus by providing also the surrounding mathematical theory. There, Newton makes use of the connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework developed by Gottfried Leibniz, who systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today. Thus, the new infinitesimal calculus allowed for precise analysis of functions within continuous domains. While Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box, Leibniz adapted the integral symbol, ∫, from the letter ∫ (long s), standing for summa (written as S*umma*; Latin for “sum” or “total”). The modern notation for the definite integral, with limits above and below the integral sign, was first used by French mathematician Joseph Fourier.[13]

“In the history of mathematics and science, few conflicts have attained the notoriety of the Newton/Leibniz dispute. … A carefully reconstructed chronology reveals that Newton had formulated the essentials of his calculus by 1666, years before Leibniz had attained the mathematical knowledge necessary to develop his own point of view on the calculus. … There is much that can never be known about such a feud. This feud is peculiar in that it erupted late, and was both sparked and carried on to a large degree by the followers of the men involved. There are scientific reasons for it (the divergences in their interpretation of “the calculus” itself; personal reasons (a history of suspicion, not only between the two principles but between each of them and other rivals; nationalism, never a negligible factor; and the bitterness associated with disputes on related matters, notably the ongoing rivalry between the Newtonian and Cartesian theories of gravity. At a personal level, Newton’s pride, suspicious character, and reluctance to publish collided with Leibniz’ naive optimism, arrogance, and his belief in “systems” as more valuable than inspiration, in a long-delayed but virulent explosion.”

– Anand Kandaswamy, in “The Newton/Leibniz Conflict in Context” (2002)

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. In 1734, Irish philosopher Bishop Berkeley memorably attacked the vanishing increments used by Newton in his critical essay ‘*The Analyst – A DISCOURSE Addressed to an Infidel Mathematician*‘, calling them “*ghosts of departed quantities*“. This was a direct attack on the foundations and principles of Infinitesimal calculus and, in particular, the notion of fluxion or infinitesimal change, which Newton and Leibniz used to develop the calculus. Calculus acquired a firmer footing with the development of limits. The first, who applied limits to rigorously formalize calculus, was German mathematician Bernhard Riemann.[14] Joseph Fourier further extended Riemann‘s concept in the context of his Fourier analysis and French mathematician Henri Lebesgue formulated a different definition of integral, founded in measure theory.[15]

At yovisto academic video search you can learn more about integral calculus in the basic lecture of MIT Prof David Jerison on ‘*Definite Integrals*‘.

**References and Further Reading:**

- [1] Biography of Bonaventura Cavallieri at MacTutor’s History of Mathematics
- [2] Biography of George Berkeley (Bishop Berkeley) at the Stanford Encyclopedia of philosophy
- [3] Biography of Bernhard Riemann at MacTutor’s History of mathematics
- [4] Biography of Evangelista Torricelli at MacTutor’s History of mathematics
- [5] Isaac Barrow at Princeton.edu
- [6] Gottfried Wilhelm Leibniz at zbMATH
- [7] Gottfried Wilhelm Leibniz at Wikidata
- [8] Timeline for Gottfried Wilhelm Leibniz, via Wikidata
- [9] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [10] Cavalieri’s Principle, SciHi Blog
- [11] Evangelista Torricelli and the Barometer, SciHi Blog
- [12] Standing on the Shoulders of Giants – Sir Isaac Newton, SciHi Blog
- [13] Joseph Fourier and the Greenhouse Effect, SciHi blog
- [14] Bernhard Riemann’s novell approaches to Geometry, SciHi Blog
- [15] Henri Léon Lebesgue and the Theory of Integration, SciHi Blog

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