The post Joseph Fourier and the Greenhouse Effect appeared first on SciHi Blog.

]]>On March 21, 1768, French mathematician and physicist **Jean Baptiste Joseph du Fourier** was born. He is probably best known for his work in thermodynamics, where he introduced the concept of the Fourier Analysis, named in honor after him. There, he claimed that every mathematical function of a variable can be expanded to a sum of sines of multiples of that variable. What people most likely don’t know is that Fourier also was the first to describe the greenhouse effect, which is responsible also for global warming.

“Profound study of nature is the most fertile source of mathematical discoveries.”

Joseph Fourier, The Analytical Theory of Heat (1878), ch.1, p.7

Jean Baptiste Joseph Fourier was born on March 21, 1768, in a modest family at Auxerre, France, as the son of a tailor. Orphaned already at age nine, Fourier was recommended to the Bishop of Auxerre, and through this introduction, he was educated by the Benvenistes of the Convent of St. Mark. While he showed an aptitude and flair for literature, this was overshadowed by mathematics, a subject he found himself really interested in at age thirteen. He proceeded to the École Royale Militaire, followed by taking admission in the Benedictine abbey of St Benoit-sur-Loire to prepare for priesthood. However, simultaneously he achieved first merits in mathematics and became rather uncertain, whether to continue his efforts towards priesthood. Military commissions in the scientific corps of the army on the other hand were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics at the Benedictine college, École Royale Militaire. Now, he also became involved in politics by taking a prominent part in his own district in promoting the French Revolution, serving on the local Revolutionary Committee. He was imprisoned briefly during the Terror in 1794, but in 1795 was appointed to study at the École Normale Supérieure, a teacher-training school set up for training teachers, and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique.

Fourier went with Napoleon Bonaparte on his Egyptian expedition in 1798 as scientific advisor, and was made governor of Lower Egypt and secretary of the Institut d’Égypte. Through the help of Fourier, Napoleon set up French type political institutions and administration. Cut off from France by the English fleet, Fourier organized the workshops on which the French army had to rely for their munitions of war. He also contributed several mathematical papers to the Egyptian Institute which Napoleon founded at Cairo, with a view of weakening English influence in the East. After the British victories and the capitulation of the French in 1801, Fourier returned to France with the remains of the expeditionary force and resumed his post as Professor of Analysis at the École Polytechnique. However Napoleon had other ideas about how Fourier might serve him. He wrote

.

.. the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place.

Fourier was not happy at the prospect of leaving the academic world in Paris, but could not refuse Napoleon’s request. He went to Grenoble where his duties as Prefect were manifold. His two greatest achievements in this administrative position were overseeing the operation to drain the swamps of Bourgoin and supervising the construction of a new highway from Grenoble to Turin. He also spent much time contributing to the monumental ‘*Description de l’Égypte*‘ which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir *On the Propagation of Heat in Solid Bodies*. Besides the derivation of the equations, it contained a solution approach using Fourier series.

The Fourier series is the series development of a periodic, continuous function in sections into a series of functions consisting of sine and cosine functions. Fourier now claimed in his work *Théorie analytique de la chaleur* that such series developments existed for all functions. This claim was initially rejected by leading mathematicians such as Cauchy [5] and Abel.[6] Based on Newton’s law of cooling, Fourier interpreted that the flow of heat between two adjacent molecules is directly proportional to the extremely small difference of their temperatures. Somehow, Fourier was obsessed with heat, keeping his rooms uncomfortably hot for visitors, while also wearing a heavy coat himself. Some as it is said trace back this eccentricity to his 3 years in Egypt.

In 1814, Fourier was placed in a tricky position, when Napoleon abdicated and set out for Elba with every likelihood of passing southward through Grenoble. To greet his old master would jeopardize his standing with the new king Louis XVIII and thus, Fourier influenced the choice of a changed route and kept his job. Unfortunately, Napoleon reappeared in France again in the very next year, this time marching north through Grenoble where he fired Fourier. However, 3 days later Fourier was appointed Prefect of the Rhone, thus surviving two changes of regime – but of course only for 100 days before the king was back in control and Napoleon was on his way to St. Helena, never to return.

Finally, Fourier moved back to Paris and back to enter scientific life again, being elected to the Académie des Sciences in 1817. In 1823, he became its permanent secretary and in 1826 also for the Académie Francaise. But, Fourier’s health started deteriorating in 1830. While he had already experienced attacks of aneurism of the heart when he was in Egypt and Grenoble, it was in Paris that the problem of suffocation became worse. A fall from the stairs on May 4, 1830, further aggravated the malady and a few days later, on May 16, 1830 Fourier passed away.

One of his lesser known legacies is the discovery of the greenhouse effect. In the 1820s Fourier calculated that an object the size of the Earth, and at its distance from the Sun, should be considerably colder than the planet actually is if warmed by only the effects of incoming solar radiation. He examined various possible sources of the additional observed heat. But, while he ultimately suggested that interstellar radiation might be responsible for a large portion of the additional warmth, his alternative consideration of the possibility that the Earth’s atmosphere might act as an insulator of some kind is widely recognized as the first proposal of what is now known as the greenhouse effect. Fourier referred to an experiment by Saussure that lined a vase with blackened cork. He used several panes of transparent glass in the corks, separated by sections of air. The midday sunlight could penetrate through the glass panes at the top of the vase. The temperature has been raised in the interiors of this unit. Fourier came to the conclusion that gases in the atmosphere would have to form a stable barrier like the glass panes for this.

Also, when Fourier returned from Egypt in 1801 with many artifacts found on the Napoleon expedition, he also had an ink pressed copy of the Rosetta Stone with him, which he happened to introduce to Jean Francois Champollion,[4] who should become the man who deciphered the Egyptian hieroglyphs with the help of the Rosetta stone. But this is already another story…..

At yovisto academic video search you can learn more about the Fourier series in mathematics with the lecture of Prof. Gilbert Strang from MIT.

**References and Further Reading: **

- [1] Ronald N. Bracewell: The Fourier Transform and Its Application, McGraw-Hill, 1986,pp 462-464. (also available via swarthmore.edu
- [2] Jean Baptiste Joseph Fourier at MacTutor’s History of Mathematics
- [3] Joseph Fourier” at thefamouspeople.com
- [4] Cracking the Code – Champollion and the Rosetta Stone, SciHi Blog, July 15, 2012.
- [5] Augustin-Louis Cauchy and the Rigor of Analysis, SciHi Blog, August 21, 2013.
- [6] The Short but Influential Life of Niels Henrik Abel, SciHi Blog, April 6, 2013.
- [7] Joseph Fourier at zbMATH
- [8] Joseph Fourier at Mathematics Genealogy Project
- [9] Joseph Fourier at Wikidata
- [10] O’Connor, John J.; Robertson, Edmund F.,
*“Joseph Fourier“*, MacTutor History of Mathematics archive, University of St Andrews. - [11] Joseph Fourier: Memoir on the temperature of the earth and planetary spaces (1827)
- [12] Timeline for Joseph Fourier, via Wikidata

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]]>The post The Life and Work of Philippe de La Hire appeared first on SciHi Blog.

]]>On March 18, 1640, French mathematician, astronomer, architect, and key figure in the Académie royale des sciences **Philippe de La Hire** was born.

Philippe de La Hire was educated as an artist and became skilled in drawing and painting early. It is believed that de La Hire received no formal education in an official school even though his father was probably teaching him at home. At the age of 16, Philippe was fully committed to becoming a professional artist and made plans to visit Italy. One reason for the journey was his poor health, which he hoped to improve. The other reason was to improve his art because in his early years, his father Laurent de La Hire had given him a love of Italian art. Starting from 1660, the young artist spent about four years attempting to develop his artistic skills and learning geometry. Soon, de La Hire realized that he enjoyed mathematics even more than painting and he began to focus more and more on geometry.

By then, La Hire befriended Abraham Bosse, with whom he could share both artistic and mathematical interests. Influenced by Bosse’s work, La Hire began working on conic sections, which he published in 1672. The publication was titled *Observations sur les Points d’Attouchement de Trois Lignes Droites qui touchent la Section d’un Cone* and it was followed by his famous treatise Nouvelle méthode en géometrie pour les sections des superficies coniques et cylindriques in 1673. According to Taton, the *Nouvelle méthode is a comprehensive study of conic sections by means of the projective approach, based on a homology which permits the deduction of the conic sections under examination from a particular circle.*

In his method, according to Taton, La Hire provided an exposition of the properties of conic sections. He began with their focal definitions and applied Cartesian analytic geometry the study of equations and the solution of indeterminate problems. He also displayed the Cartesian method for solving certain types of equations by intersections of curves. Although not a work of great originality, it summarises the progress achieved in analytical geometry during half a century and contained some interesting ideas, among them the possible extension of space to more than three dimensions.”

In January 1678, La Hire was elected to the Académie des Sciences due to his publications in geometry. It was a great honor for the artist and scientist and he was assigned by Jean-Baptiste Colbert,[6] the French Minister of Finance, to assist Jean Picard in the surveying work in order to create more accurate maps of France.[4] He was the first to prove that the hypotrochoids of the Cardanic Circles are all ellipses and developed a circle of ellipses from this.

Together, La Hire and Picard worked in Brittany in 1679 and in Guyenne in 1880. La Hire then went, without Jean Picard, to survey around Calais and Dunkirk in 1681 and the coast of Provence in 1682. By that time, La Hire’s work for the Academy was closely linked to the Paris Observatory which had been founded largely due to Colbert. Also in that period, La Hire was appointed to the chair of mathematics at the Collège Royale. He was known to be a great teacher, who put much work in his lectures. As a mechanic of epicyclic gear theory, he continued the work of Christian Huygens.[5] In France, it is attributed the hypocycloidal train whose inner wheel has a radius half of the basic wheel, the center of the rolling one describing a periodic translation. In 1702, he is the first to explain the movement of the rockets by the force of the expanded air acting on all the interior of the rocket except the lower orifice.

Also, La Hire lectured his son in the same way he was educated by his father. His son, eventually joined his father’s teaching activities, which included the fields of mathematics, astronomy, mechanics, hydrostatics, dioptrics, and navigation. Gabriel-Philippe La Hire became the youngest member of the Academy in the seventeenth century.

During his career, Philippe de La Hire contributed to many fields of science, even though he always preferred geometry. He published a comprehensive work on conic sections which contained a description of Desargues’ projective geometry in 1685. He calculated the length of the cardioid and wrote about the cycloid, the epicycloid, the conchoid and quatratures. In astronomy he installed the first transit instrument in the Paris Observatory. He also produced tables giving the movements of the Sun, Moon and the planets which he published in 1687, publishing further such tables in 1702.

At yovisto academic video search, you may learn more about ‘*The beautiful math that links coral, crochet and hyperbolic geometry*‘ in a video lecture by Margaret Wertheim.

**References and Further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson:
*Philippe de La Hire.*In:*MacTutor History of Mathematics archive.* - [2] Short biography of Philippe de La Hire at Structurae
- [3] Philippe de La Hire in the The Biographical Encyclopedia of Astronomers
- [4] Jean Picard and his Love for Accuracy, SciHi Blog, July 21, 2014.
- [5] Christiaan Huygens and the Pocket Watch, SciHi Blog, October 4, 2015.
- [6] The Invention of Financial Politics by Jean-Baptiste Colbert, SciHi Blog
- [7] Philippe de la Hire at Wikidata
- [8] Timeline of 17th Century French Mathematicians, via DBpedia and Wikidata

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]]>The post Frank P. Ramsey and the Ramsey Theory appeared first on SciHi Blog.

]]>On February 22, 1903, precocious British philosopher, mathematician and economist **Frank Plumpton Ramsey** was born. Although he died already at age 26, he had made significant contributions to logic, philosophy of mathematics, philosophy of language and decision theory. He remains noted for his Ramsey Theory, a mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large.

“The first problem I propose to tackle is this: how much of its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows.

The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment.”

– Frank P. Ramsey, “A Mathematical Theory of Saving”, The Economic Journal, Vol. 38, No. 152 (Dec., 1928)

Frank P. Ramsey was born as the eldest of four siblings in Cambridge where his father Arthur Stanley Ramsey, also a mathematician, was President of Magdalene College. His brother Michael Ramsey later should become Archbishop of Canterbury. Ramsey entered Winchester College in 1915 and later returned to Cambridge to study mathematics at Trinity College. At Trinity College, Ramsey became a student to John Maynard Keynes.[3] He became a senior scholar in 1921 and graduated as a Wrangler in the Mathematical Tripos of 1923. Then, Ramsey went to Vienna for a short while, returning to Cambridge where he was elected a fellow of King’s College Cambridge in 1924, which was rather unusual, because in fact Ramsey was only the second person ever to be elected to a fellowship at King’s College, not having previously studied at King’s.[1]

In 1926 he was appointed as a university lecturer in mathematics and he later became a Director of Studies in Mathematics at King’s College. It was a short career, for sadly Ramsey died at the beginning of 1930. Suffering from chronic liver problems, Ramsey developed jaundice after an abdominal operation and died on 19 January 1930 at Guy’s Hospital in London at the age of 26. However, in the short time during which he lectured at Cambridge he had already established himself as an outstanding lecturer. He published his first major work *The Foundations of Mathematics* in 1925, in which he accepted the claim by Russell [4] and Whitehead made in the *Principia** Mathematica* that mathematics is a part of logic.

Already at the age of 19, Ramsey was able to make the first draft of the translation of the German text of Ludwig Wittgenstein’s *Tractatus Logico Philosophicus*,[5] Wittgenstein’s seminal work that aimed to identify the relationship between language and reality and to define the limits of science. Ramsey was very impressed by Wittgenstein’s work and after graduating in 1923 he made a journey to Austria to visit Wittgenstein. For two weeks Ramsey discussed the difficulties he was facing in understanding the *Tractatus*. Wittgenstein made some corrections to the English translation in Ramsey’s copy and some annotations and changes to the German text that subsequently appeared in the second edition in 1933.

In his second paper on mathematics *On a problem of formal logic*, which was published in the *Proceedings of the London Mathematical Society* in 1930, he examined methods for determining the consistency of a logical formula and included some theorems on combinatorics which have led to the study of a whole new area of mathematics called Ramsey theory. The combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate calculus. Ramsey theory studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: “*how many elements of some structure must there be to guarantee that a particular property will hold?”*

Ramsey made a systematic attempt to base the mathematical theory of probability on the notion of partial belief. This work on probability, and also important work on economics, came about mainly because Ramsey was a close friend of John Meynard Keynes.[3] Being a friend of Keynes did not stop Ramsey attacking Keynes’ work, however, and in *Truth and probability* (1926) he argues against Keynes’ ideas of an a priori inductive logic. Ramsey’s arguments convinced Keynes who then abandoned his own ideas.

In economics, Ramsey wrote two papers A contribution to the theory of taxation and A mathematical theory of saving. These would lead to important new areas in the subject. In *A mathematical theory of saving*, published in *The Economic Journal* in late 1928, Ramsey aimed to determine the optimal amount an economy should invest (save) rather than consume so as to maximize future utility, or in Ramsey’s words “*how much of its income should a nation save?*” Keynes described the article as “one of the most remarkable contributions to mathematical economics ever made. The Ramsey model is today acknowledged as the starting point for optimal accumulation theory although its importance was not recognized until many years after its first publication.

Ramsey suffered an attack of jaundice and was taken to Guy’s Hospital in London for an operation. He died following the operation, on January 19, 1930, at age 26.

At yovisto academic video search, you can learn more about the work of Frank P. Ramsey in the lecture “A Dictator Theorem of Belief Revision Derived from Arrow’s Theorem” at LMU Munich, where Hannes Leitgeb discusses the Ramsey test for conditionals.

**References and Further Reading: **

- [1] O’Connor, John J.; Robertson, Edmund F., “Frank P. Ramsey“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] D.H.Mellor: Cambridge Philosophers I: F. P. Ramsey, in Philosophy 70, 243-262,1995
- [3] John Maynard Keynes and his General Theory of Employment, SciHi Blog, June 5, 2013.
- [4] The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist, SciHi Blog, July 11, 2012.
- [5] The Philosophy of Ludwig Wittgenstein, SciHi Blog, April 26, 2016.
- [6] Frank P. Ramsey at Wikidata
- [7] Frank P. Ramsey at Reasonator
- [8] Frank P. Ramsey at zbMATH
- [9] Frank P. Ramsey at Mathematics Genealogy Project
- [10] Ramsey, F.P. (1931),
*The Foundations of Mathematics, and other Essays*, (ed.) R. B. Braithwaite - [11] Ramsey, F.P. (1929). “On a Problem in Formal Logic” (PDF).
*Proc. London Math. Soc*.**30**: 264–286. - [12] Ramsey, F.P. (1927). “Facts and Propositions” (PDF).
*Aristotelian Society Supplementary*.**7**: 153–170. - [13] Better than the Stars/Frank Ramsey: a biography a 1978 BBC radio portrait of Ramsey and a 1995 article derived from it, both by David Hugh Mellor.
- [14] Timeline of Frank P. Ramsey, via Wikidata

The post Frank P. Ramsey and the Ramsey Theory appeared first on SciHi Blog.

]]>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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