The post Pierre de Fermat and his Last Problem appeared first on SciHi Blog.

]]>On January 12, 1665, French lawyer and amateur mathematician **Pierre de Fermat**, famous for his research in number theory, analytical geometry and probability theory, passed away. He is best known for Fermat’s Last Theorem, which he described in a note at the margin of a copy of Diophantus’ *Arithmetica*.[4]

Born into a wealthy French family, Pierre de Fermat grew up in Beaumont de Lomagne and later attended the University of Toulouse. After moving to Bordeaux a few years later, de Fermat began his first mathematical studies and published his first works in 1629. His influences during this period were Jean de Beaugrand, Ètienne d’Espagnet and François Viète.[5] It was also during these years, when the mathematician published his works on maxima and minima. Pierre de Fermat earned a degree in civil law in 1626 and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held until the end of his lifetime.

The hobby-mathematician pioneered in analytic geometry, which he noted in a manuscript published posthumously in 1679 in “*Varia opera mathematica*“. He was able to increase his good reputation through evaluating the integral of power functions, which influenced the work of Newton and Leibniz. Fermat’s contributions to number theory were groundbreaking. While studying Pell numbers, he discovered the ‘little theorem‘ and invented a factorization method and the proof technique of infinite descent that he used for proving this theorem for n = 4. Further theorems he discovered were the two-square theorem as well as the polygonal number theorem.

Fermat’s famous ‘*Last Theorem*‘ saying that no three positive integers a, b, and c can satisfy the equation a^{n}+b^{n}=c^{n} for any integer value of n greater than two, was not published by Fermat himself. His son Samuel, who collected all of his father’s works did so to honor his father’s works. However, the real problem was the theorem’s proof since Fermat himself only wrote

“

I have discovered a truly remarkable proof which this margin is too small to contain.“

This little note was written around 1630, when the hobby-mathematician studied Diophantus’ ‘*Arithmetica*‘ but lacked of a real proof, which numerous mathematicians used as a challenge to find in the subsequent centuries. The French mathematician, physicist, and philosopher Sophie Germain proved the theorem for a special case for all primes less than 100 [7] and Ernst Kummer proved it for regular primes in the middle of the 19th century [7].

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat’s Last Theorem. Finally a proof for all *n* came closer in 1984, when Gerhard Frey suggested proving the theorem with a proof of the modularity theorem for elliptic curves. And in 1995 Andrew Wiles succeeded in the final proof building on the work of Ken Ribet, and assisted by Richard Taylor. In fact Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997. The challenge, which de Fermat gave these later mathematicians took three centuries to be proven and was therefore listed in the ‘Guinness Book of World Records’ for the most difficult mathematical problems.

Fermat’s Last problem also has many references in popular culture. The one I remember comes from Star Trek Next Generation. In one episode, entiteled ‘*The Royal*‘, Commander Riker visits Captain Jean-Luc Picard in his ready room to report only to find Picard puzzling over Fermat’s last theorem. Picard’s interest in this theorem goes beyond the difficulty of the puzzle; he also feels humbled that despite their advanced technology of the 23rd century, they are still unable to solve a problem set forth by a man who even had no computer. Obviously, this episode was shot before Wiles’ proof has been published.

Pierre de Fermat passed away on January 12, 1665 and depicted one of the most influential mathematicians of the 17th century, especially in the field of number theory along with the famous René Descartes.

At yovisto, you may enjoy the video lecture on ‘Number Theory’ by Steven Skiena at Hong Kong University of Science and Technology.

**Refenences and Further Reading:**

- [1] The Proof of Fermat’s Last Theorem by R. Taylor and A.Wiles by Gerd Faltings [PDF]
- [2] Pierre de Fermat biography at robertnowlan.com
- [3] Fermat’s Last Theorem
- [4] Diophantos of Alexandria – the father of Algebra, SciHi Blog, April 2, 2016.
- [5] Francois Viète and his New Algebra, SciHi Blog, February 23, 2016.
- [6] Sophie Germain and the Chladni Experiment, SciHi Blog, June 27, 2014.
- [7] Ernst Kummer and his Achievements in Mathematics, SciHi Blog, January 29, 2015.
- [8] Standing on the Shoulders of Giants – Sir Isaac Newton, SciHi Blog, January 4, 2018.
- [9] Pierre de Fermat at zbMATH
- [10] O’Connor, John J.; Robertson, Edmund F., “Pierre de Fermat“, MacTutor History of Mathematics archive, University of St Andrews.
- [11] Timeline of things named after Pierre de Fermat, via Wikidata

**Related Articles in the yovisto Blog:**

- George Boole – The Founder of Modern Logics
- The bustling Life and Publications of Mathematician Paul Erdös
- How to Calculate Fortune – Jakob Bernoulli
- David Hilbert’s 23 Problems
- Ernst Haeckel and the Phyletic Museum

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]]>The post Donald Knuth and the Art of Programming appeared first on SciHi Blog.

]]>photo: October 25, 2005 by Jacob Appelbaum

On January 10, 1938, computer scientist **Donald Knuth**, developer of the seminal computer science textbooks ‘*The Art of Computer Programming*‘, was born. He is also widely known for his development of the TeX typesetting framework and the *METAFONT* font definition language. Actually, Donald Knuth is one of my personal heroes in computer science. The very day I started to study this subject, his textbooks had already become a sort of ‘holy bible’ when it comes to algorithms and esp. the analysis of algorithms, i.e. the very heart of theoretical computer science. However, about the person behind the seminal book, I knew almost next to nothing…

Donald Ervin Knuth was born in Milwaukee, Wisconsin, where his father owned a small printing business and taught bookkeeping at Milwaukee Lutheran High School, where Donald enrolled, earning achievement awards. He applied his intelligence in unconventional ways, winning a contest set up by the confectionary manufacturer Ziegler when he was in eighth grade by finding over 4,500 words that could be formed from the letters in “Ziegler’s Giant Bar”. However, the judges had only about 2,500 words on their master list. This won him a television set for his school and a candy bar for everyone in his class.

At high school Knuth’s interests were more directed towards music than they were to mathematics. His musical interests involved both playing and composing music and he decided at that stage that he would study music after graduating from high school. Knuth played the saxophone, and later the tuba, in his school band. However, in 1956 he graduated from High School with the highest grade point average that anyone had ever achieved at his school. He decided to take physics as his major at Case Institute of Technology (now part of Case Western Reserve University), where he was introduced to the IBM 650 computer, one of the early mainframes. After reading the computer’s manual, he decided to rewrite the assembler and compiler code for the machine used in his school, because he believed he could do it better.

In 1958, Knuth constructed a program based on the value of each player that could help his school basketball team win the league. This was so novel a proposition at the time that it got picked up and published by Newsweek and also covered by Walter Cronkite on the CBS Evening News. Knuth was one of the founding editors of the Engineering and Science Review, which won a national award as best technical magazine in 1959. He then switched from physics to mathematics, and in 1960 he received his bachelor of science degree, simultaneously receiving his master of science degree by a special award of the faculty who considered his work outstanding. Knuth was awarded two Fellowships, a Woodrow Wilson Fellowship and a National Foundation Fellowship in the year of his graduation.[6]

In the autumn of 1960 Knuth entered the California Institute of Technology and, in June 1963, he was awarded a Ph.D. in mathematics for his thesis ‘*Finite semifields and projective planes*‘. In fact in addition to the work for his doctorate in mathematics, Knuth had from 1960 begun to put his very considerable computing expertise to uses other than writing papers becoming a software development consultant to the Burroughs Corporation in Pasadena, California. Besides, knowledge of his computing expertise was so well established by 1962 that, although he was still a doctoral student at the time, Addison-Wesley approached him and asked him to write a text on compilers. He began to work at CalTech as associate professor and the commission from Addyson-Wesley turned out into the writing of his seminal multivolume book ‘*The Art of Computer Programming*‘. This work was originally planned to be a single book, and then planned as a six- and then seven-volume series. In 1968, just before he published the first volume, Knuth was appointed as Professor of Computer Science at Stanford University. After producing the third volume of his book series in 1976, he expressed such frustration with the nascent state of the then newly-developed electronic publishing tools (especially those that provided input to phototypesetters) that he took time out to work on typesetting and created the *TeX* and *METAFONT* tools. As of 2012, the first three volumes and part one of volume four of his series have been published.

Knuth was appointed Fletcher Jones Professor of Computer Science in 1977 and in 1990 he was named Professor of The Art of Computer Programming. In 1993 he became Professor Emeritus at Stanford University and continued to live on the University Campus.[5] Knuth has made many contributions to mathematics and computing. One particular contribution we should mention is the Knuth-Bendix algorithm, one of the fundamental algorithms for computing with algebraic structures, particularly with groups and semigroups.

TeX has changed the technology of mathematics and science publishing since it enables mathematicians and scientists to produce the highest quality of printing of mathematical articles yet this can be achieved simply using a home computer. However, it has not only changed the way that mathematical and scientific articles are published but also in the way that they are communicated.[5]

At yovisto, you can watch Prof. Donald Knuth himself in his 17th annual Christmas Tree lecture at Stanford University about ‘Bayesian trees and BDDs‘.

**References and Further Reading:**

- [1] Donald Knuth at Mac Tutor’s History of Mathematics
- [2] Donald Knuth at wikipedia
- [3] Donald E. Knuth,
*The Art of Computer Programming*, Volumes 1–4, Addison-Wesley Professional - [4] Donald Knuth at zbMATH
- [5] O’Connor, John J.; Robertson, Edmund F., “Donald Knuth“, MacTutor History of Mathematics archive, University of St Andrews.
- [6] Donald Knuth at Wikidata
- [7] Scholia entry for Donald Knuth
- [8] Timeline of works by Donald Knuth, via Wikidata

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]]>The post Standing on the Shoulders of Giants – Sir Isaac Newton appeared first on SciHi Blog.

]]>Portrait by Sir Godfrey Kneller (1689)

On January 4, 1643 [N.S.] (25 December 1642 [O.S.]), **Sir Isaac Newton**, famous physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, was born. With his *Principia* Newton laid the foundation of modern classical mechanics. Besides he constructed the very first reflecting telescope and independent of Gottfried Wilhelm Leibniz developed differential and integral calculus [10].

“We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”

Isaac Newton, Philosophiae Naturalis Principia Mathematica (1687), “Rules of Reasoning in Philosophy” : Rule I

On January 4, 1643, (December 25, 1642 according to the old Iulian calendar) Isaac Newton was born in the hamlet of Woolsthorpe, England, the only son of a prosperous local farmer, also named Isaac Newton. Young Isaac never knew his father, who died three months before he was born. A premature baby born tiny and weak, Isaac was not expected to survive. When he was three, his mother remarried a minister, Barnabas Smith, and went to live with him, leaving young Isaac with his maternal grandmother. At age twelve, Isaac Newton was reunited with his mother after her second husband died. Although he had been enrolled at the King’s School, Grantham, England, his mother pulled him out of school, for her plan was to make him a farmer and have him tend the farm. But, Isaac failed miserably for he found farming rather monotonous. Soon he was returned to King’s School to finish his basic education.

The turning point in Newton’s life came in June 1661 when he left for Cambridge University, the outstanding center of learning in these days. He was older than most of his fellow students but, despite the fact that his mother was financially well off, he entered as a sizar. A sizar at Cambridge was a student who received an allowance toward college expenses in exchange for acting as a servant to other students. There is certainly some ambiguity in his position as a sizar, for he seems to have associated with “better class” students rather than other sizars [13].

In 1664 Isaac Barrow, Lucasian Professor of Mathematics at Cambridge, examined Newton’s understanding of Euclid and found it sorely lacking. This was partly because Newton was rather occupied with his private study of the works of René Descartes [8], Pierre Gassendi [9], Thomas Hobbes [3], and other major figures of the scientific revolution. The mechanics of the Copernican astronomy of Galileo attracted him and he also studied Kepler’s Optics. He recorded his thoughts in a book which he entitled *Quaestiones Quaedam Philosophicae* (*Certain Philosophical Questions*).[13] In 1665 Newton took his bachelor’s degree. Since the university was closed for the next two years because of plague, Newton returned to Woolsthorpe in midyear, where in the following 18 months, he made a series of original contributions to science. In mathematics Newton conceived his ‘*method of fluxions*‘ (infinitesimal calculus), laid the foundations for his theory of light and color, and achieved significant insight into the problem of planetary motion, insights that eventually led to the publication of his *Principia* in 1687.

The ‘method of fluxions’, as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions.[13]

In 1667, Newton returned to Cambridge, where in the next year he became a senior fellow upon taking his master of arts degree, and in 1669, he succeeded Isaac Barrow as Lucasian Professor of Mathematics. Incredible, if you take into account that Newton was barely 27 years of age. He had reached the conclusion during the two plague years that white light is not a simple entity. Every scientist since Aristotle had believed that white light was a basic single entity, but the chromatic aberration in a telescope lens convinced Newton otherwise. When he passed a thin beam of sunlight through a glass prism Newton noted the spectrum of colours that was formed. He argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a different spectral colour. Newton was led by this reasoning to the erroneous conclusion that telescopes using refracting lenses would always suffer chromatic aberration. He therefore proposed and constructed a reflecting telescope. [13] At Cambridge, Newton was able to organize the results of his optical researches and in 1672, shortly after his election to the Royal Society after donating his reflecting telescope, he communicated his first public paper, a brilliant but no less controversial study on the nature of color.

“Hypotheses non fingo”. (I frame no hypotheses.),

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Third edition

Due to a dispute with his fellow scientist Robert Hooke, who claimed that Newton had stolen some of his optical results, Newton turned in on himself and away from the Royal Society which he associated with Hooke as one of its leaders. He even delayed the publication of a full account of his optical researches until after the death of Hooke in 1703. Newton’s *Opticks* appeared in 1704. In 1687, with the support of his friend the astronomer Edmond Halley [6], Newton published his single greatest work, the ‘*Philosophiae Naturalis Principia Mathematica*‘, in which he showed how a universal force, gravity, applied to all objects in all parts of the universe. The *Principia* is recognised as the greatest scientific book ever written. Newton analysed the motion of bodies in resisting and non-resisting media under the action of centripetal forces. The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another.[13]

In 1689, Newton was elected member of parliament for Cambridge University and in 1696, by the support of his friend and ex-student, Charles Montagu, 1st Earl of Halifax, Newton was appointed warden of the Royal Mint, settling in London. He took his duties at the Mint very seriously and campaigned against corruption and inefficiency within the organisation. As a scholar, Newton held court in the fashionable London coffee houses, surrounded by his acolytes, for whom the term Newtonians was originally minted, handing out unpublished manuscripts to the favoured few for their perusal and edification [15]. In 1703, he was elected president of the Royal Society and knighthood followed in 1705. Newton was referred to being a rather difficult man, prone to depression and often involved in bitter arguments with other scientists, but by the early 1700s he was the dominant figure in British and European science. He died on 31 March 1727 and was buried in Westminster Abbey.

At yovisto you might learn more about Isaac Newton’s laws of physics in the famous MIT lecture of Prof. Walter Lewin on classical mechanics and Newton’s laws.

**References and Further Reading:**

- [1] Isaac Newotn ond his famous Principia, at yovisto blog
- [2] Isaac Newton at biography.com
- [3] Man is Man’s Wolf – Thomas Hobbes and his Leviathan, SciHi Blog, December4, 2012.
- [4] Dr. Robert A. Hatch: Sir Isaac Newton
- [5] Isaac Newton at BBC History
- [6] Edmond Halley besides the Eponymous Comet, SciHi Blog, November 8, 2015.
- [7] Sir Edmond Halley and his Famous Comet, SciHi Blog, November 8. 2012.
- [8] Cogito Ergo Sum – René Descartes, SciHi Blog, March 31, 2013.
- [9] Pierre Gassendi and his Trials to reconcile Epicurean atomism with Christianity, January 22, 2015.
- [10] Leibniz and the Integral Calculus, SciHi Blog, November 11, 2013.
- [11] Sir Isaac Newton at Wikidata
- [12] Isaac Newton at zbMATH
- [13] O’Connor, John J.; Robertson, Edmund F., “Sir Isaac Newton“, MacTutor History of Mathematics archive, University of St Andrews.
- [14] Isaac Newton at Mathematics Genealogy Project
- [15] Christmas Trilogy 2017 Part 1: Isaac the Imperator, at The Renaissance Mathematicus, Dec 25, 2017.
- [16] Timeline for Isaac Newton, via Wikidata

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]]>The post The Celestial Mechanics of Anders Johann Lexell appeared first on SciHi Blog.

]]>On December 24, 1740, Finnish-Swedish astronomer, mathematician, and physicist Anders Johann Lexell was born. Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in his honour. La Grande Encyclopédie states that he was the prominent mathematician of his time who contributed to spherical trigonometry with new and interesting solutions, which he took as a basis for his research of comet and planet motion. His name was given to a theorem of spherical triangles.

Anders Johan Lexell studied at the University of Åbo and earned his Doctor of Philosophy degree with a dissertation on *Aphorismi mathematico-physici*. Later on, Lexell moved to Uppsala and began working at Uppsala University as a mathematics lecturer. In 1766, Lexell became professor of mathematics at the Uppsala Nautical School.

During the 1760s, Catherine the Great ascended to the Russian throne and started the politics of enlightened absolutism. As she was aware of the importance of science, she ordered to offer Leonhard Euler to “state his conditions, as soon as he moves to St. Petersburg without delay”. After his return to Russia, Leonard Euler suggested that the director of the Russian Academy of Science should invite mathematics professor Anders Johan Lexell to study mathematics and its application to astronomy, especially spherical geometry.

Lexell intended to become a member of the Russian Academy of Sciences. In order to be admitted he successfully wrote a paper on integral calculus titled *Methodus integrandi nonnulis aequationum exemplis illustrata*. Lexell moved to St. Petersburg and first became familiar with the astronomical instruments that would be used in the observations of the transit of Venus. He participated in observing the 1769 transit at St. Petersburg together with Christian Mayer.

In 1775, Anders Lexell was appointed to a chair of the mathematics department at the University of Åbo with permission to stay at St. Petersburg for another three years to finish his work there. His permission was later prolonged for two more years. During the 1780s, Lexell departed St. Petersburg for a study trip. He arrived in Berlin, and traveled to Potsdam, seeking in vain for an audience with King Frederick II. He then traveled to Bavaria visiting Leipzig, Göttingen, and Mannheim on the way. Later, Lexell traveled to Strasbourg and then to Paris, where he spent the winter. In 1781, he moved to London and after a few months, he left for Belgium and the Netherlands, returning to Germany in fall. Lexell further traveled to Sweden and returned to St. Peterburg in December. During his trip, Anders Lexell wrote numerous letters to Johann Euler with descriptions of places and people he met.

Anders Lexell is best known for his efforts in the field of celestial mechanics. His first work at the Russian Academy of Sciences was to analyse data collected from the observation of the 1769 transit of Venus. He further calculated the parallax of the Sun, and published his work in 1772. During the next decade, he computed the orbits of all the newly discovered comets, among them the comet which Charles Messier discovered in 1770. Lexell computed its orbit, showed that the comet had had a much larger perihelion before the encounter with Jupiter in 1767 and predicted that after encountering Jupiter again in 1779 it would be altogether expelled from the inner Solar System. This comet was later named Lexell’s Comet.

Further, Anders Lexell calculated the orbit of Uranus and he proved that it was a planet, and not a comet. During his time in Europe, Lexell already made preliminary computations and after returning to Russia, he was able to make more precise calculations. However, due to the long orbital period it was still not enough data to prove that the orbit was not parabolic. Still, Lexell found data gathered by Christian Mayer in Pisces that was not in the Flamsteed catalogues. Lexell presumed that it was an earlier sighting of the same astronomical object and using this data he calculated the exact orbit, which proved to be elliptical, and proved that the new object was actually a planet. Anders Lexell also estimated the planet’s size more precisely than his contemporaries using Mars that was in the vicinity of the new planet at that time.

**References and Further Reading:**

- Anders Johann Lexell at MacTutor History of Mathematics Archive
- Anders Johann Lexell at the SEDS Messier Database
- Anders Johann Lexell at Wikidata

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]]>The post Sophus Lie and the Lie Theory appeared first on SciHi Blog.

]]>On December 17, 1842, Norwegian mathematician **Sophus Lie** was born. Lie largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He made significant contributions to the theories of algebraic invariants, continuous groups of transformations and differential equations. Lie groups and Lie algebras are named after him.

Marius Sophus Lie was born in Nordfjordeid, near Bergen, Norway, to his father Johann Lie, pastor in Moss am Kristianiafjord, and his wife Mette Stabell, the daughter of a chief customs officer. His parents had six children and Sophus was the youngest of the six. Sophus first attended school in the town of Moss, which is a port in south-eastern Norway, on the eastern side of the Oslo Fjord. In 1857 he entered Nissen’s Private Latin School in Christiania (today’s Oslo), where he decided to take up a military career, but his eyesight was not sufficiently good so he gave up the idea and entered University of Christiania in 1859. Lie studied natural sciences from 1859 to 1865 and in 1862 he attended lectures on group theory with Peter Ludwig Mejdell Sylow, a Norwegian mathematician who wrote fundamental work on group theory. In 1865 he passed the examination as a real teacher in 1865 and was initially indecisive about his further career. The one thing he knew he wanted was an academic career and he thought for a while that astronomy might be the right topic. He learnt some mechanics, wondered whether botany or zoology or physics might be the right subjects and in general became rather confused. However, there are signs that from 1866 he began to read more and more mathematics and the library records in the University of Christiania show clearly that his interests were steadily turning in that direction.[1]

His first mathematical work, *Repräsentation der Imaginären der Plangeometrie*, was published, in 1869, by the Academy of Sciences in Christiania and also by *Crelle’s Journal*. That same year he received a scholarship and traveled to Berlin, where he stayed from September to February 1870. In Berlin he met Kronecker, Kummer and Weierstrass. Lie was not attracted to the style of Weierstrass’s mathematics which dominated Berlin. However, in Berlin he also met German mathematician Felix Klein and they became close friends. When he left Berlin, Lie traveled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan, known both for his foundational work in group theory and for his influential *Cours d’analyse*, and Gaston Darboux, the biographer of Henri Poincaré and editor of the *Selected Works of Joseph Fourier*. While in Paris Lie discovered contact transformations. These transformations allowed a 1-1 correspondence between lines and spheres in such a way that tangent spheres correspond to intersecting lines.[1] But with the outbreak of the 1870 Franco-Prussian War, Klein, who was Prussian, had to leave France very quickly. Lie left for Fontainebleau where after a while he was arrested under suspicion of being a German spy, an event that made him famous in Norway. He was released from prison after a month, thanks to the intervention of Darboux.

Lie obtained his PhD at the University of Christiania in 1871 with a thesis entitled *On a class of geometric transformations*. The dissertation contained ideas from his first results published in *Crelle’s Journal* and also the work on contact transformations, a special case of these transformations being a transformation which maps a line into a sphere, which he had discovered while in Paris. The next year, the Norwegian Parliament established an extraordinary professorship for him. In 1884, Although Lie was producing highly innovative mathematics, he became increasingly sad at the lack of recognition he was receiving in the mathematical world. One reason was undoubtedly his isolation in Christiania, but a second reason was that his papers were not easily understood, partly through his style of writing and partly because his geometrical intuition greatly exceeded that of other mathematicians. Klein, realising the problems, had the excellent idea of sending Friedrich Engel to Christiania to help Lie.[1] Engel, who had received his doctorate from Leipzig in 1883 having studied under Adolph Mayer writing a thesis on contact transformations, would help Lie to write his most important treatise, *Theorie der Transformationsgruppen*, published in Leipzig in three volumes from 1888 to 1893.

In 1886 Lie became professor at Leipzig as successor of Felix Klein, who had moved on to Göttingen. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. After that, he returned to his post, but over the years his anaemia progressed to the point where he decided to return to his homeland. Consequently, in 1898 he tendered his resignation in May, and left for home in September the same year. He died the following year, 1899, at age 56.

Lie founded the theory of continuous symmetry and used it to study differential equations and geometric structures. Continuous or continuous symmetry operations are, for example, shifts and rotations around arbitrary, also infinitesimal, amounts, in contrast to discrete symmetry operations such as reflections. Based on his work, he developed an algorithm for the numerical integration of differential equations (Lie-Integration) or the method of footpoint transformation. In order to investigate and apply continuous transformation groups (today called Lie groups), he linearized the transformations and investigated the infinitesimal generators. The linkage properties of the Lie group can be expressed by commutators of the generators; today, the commutator algebra of the generators is called Lie algebra. Hermann Weyl used Lie’s work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.

Lie was a foreign member of the Royal Society, as well as an honorary member of the Cambridge Philosophical Society and the London Mathematical Society, and his geometrical inquiries gained him the much-coveted honour of the Lobatchewsky prize.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F. (February 2000), “Sophus Lie“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Camille Jordan and the Cours d’Analyse. ScviHi Blog, January 5, 2017.
- [3] Felix Klein and the Klein-Bottle. SciHi Blog, April 25, 2016.
- [4] Sophus Lie at the Professorenkatalog of the University of Leipzig
- [5] Sophus Lie at the Mathematics Genealogy Project
- [6] Marius Sophus Lie at zbMATH
- [7] Sophus Lie at Wikidata
- [8] Timeline for mathematician Sophus Lie, via Wikidata

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]]>The post Edmund Gunter and his Measuring Devices appeared first on SciHi Blog.

]]>On December 10, 1626, English clergyman, mathematician, geometer and astronomer **Edmund Gunter** passed away. Gunther is best remembered for his mathematical contributions which include the invention of the Gunter’s chain, the Gunter’s quadrant, and the Gunter’s scale. In 1620, he invented the first successful analog device which he developed to calculate logarithmic tangents.

Edmund Gunter was born in Hertfordshire in 1581. Edmund attended Westminster School as a Queen’s Scholar, then entered Christ Church, Oxford on 25 January 1600. He graduated with a B.A. on 12 December 1603. Already as an undergraduate, Gunter had developed a strong interest in mathematics and in mathematical instruments. He wrote a manuscript *New Projection of the Sphere* in his final year as an undergraduate and this manuscript, which brought him to the attention of a number of leading mathematicians of the time including Henry Briggs. Continuing his studies at Oxford, Gunter was awarded an M.A. in 1606 but he remained at Oxford until 1615 when he received the divinity degree of B.D. on 23 November. Gunter was ordained and in 1615 became Rector of St George’s Church in Southwark and of St Mary Magdalen, Oxford. He held these positions in the Church until his death.[1]

Mathematics, particularly the relationship between mathematics and the real world, was the one overriding interest throughout his life.

Gunter became a friend of Henry Briggs, the first Professor of Mathematics at Gresham Colleg, and would spend much time with him at Gresham College discussing mathematical topics. In 1619, Sir Henry Savile put up money to fund Oxford University’s first two science faculties, the chairs of astronomy and geometry. Gunter applied to become professor of geometry but Savile was famous for distrusting clever people, and Gunter’s behavior annoyed him intensely. As was his habit, Gunter arrived with his sector and quadrant, and began demonstrating how they could be used to calculate the position of stars or the distance of churches, until Savile could stand it no longer. “*Doe you call this reading of Geometric?*” he burst out. “*This is mere showing of tricks, man!” and, according to a contemporary account, “dismissed him with scorne.*” He was shortly thereafter championed by the far wealthier Earl of Bridgewater, who saw to it that on 6 March 1619 Gunter was appointed professor of astronomy in Gresham College, London. This post he held till his death.

A competent but unoriginal mathematician, he had a gift for devising instruments which simplified calculations in astronomy, navigation, and surveying. With Gunter’s name are associated several useful inventions, descriptions of which are given in his treatises on the sector, cross-staff, bow, quadrant and other instruments. He contrived his sector about the year 1606, and wrote a description of it in Latin, but it was more than sixteen years afterwards before he allowed the book to appear in English. In 1620 he published his *Canon Triangulorum, or Table of Artificial Sines and Tangents*, the first published table of common logarithms of the sine and tangent functions, he introduced the terms cosine and cotangent. Its uses included the solution of plane, spherical, and nautical triangles (the last formed from rhumb, meridian, and latitude lines). With improvements, the British navy used it for two centuriesIn 1624 Gunter published a collection of his mathematical works. It was entitled *The description and use of sector, the cross-staffe, and other instruments for such as are studious of mathematical practise*. One of the most remarkable things about this book is that it was written, and published, in English not Latin. It was a manual not for cloistered university fellows but for sailors and surveyors in real world.

There is reason to believe that Gunter was the first to discover (in 1622 or 1625) that the magnetic needle does not retain the same declination in the same place at all times. By desire of James I he published in 1624 *The Description and Use of His Majesties Dials in Whitehall Garden*, the only one of his works which has not been reprinted.

Gunter’s interest in geometry led him to develop a method of sea surveying using triangulation. Linear measurements could be taken between topographical features such as corners of a field, and using triangulation the field or other area could be plotted on a plane, and its area calculated. A chain 20 metres long, with intermediate measurements indicated, was chosen for the purpose, and is called Gunter’s chain. The length of the chain chosen, 20 m, being called a chain gives a unit easily converted to area. Therefore, a parcel of 10 square chains gives 1 acre. The area of any parcel measured in chains will thereby be easily calculated.

Gunter’s quadrant is an instrument made of wood, brass or other substance, containing a kind of stereographic projection of the sphere on the plane of the equinoctial, the eye being supposed to be placed in one of the poles, so that the tropic, ecliptic, and horizon form the arcs of circles, but the hour circles are other curves, drawn by means of several altitudes of the sun for some particular latitude every year. This instrument is used to find the hour of the day, the sun’s azimuth, etc., and other common problems of the sphere or globe, and also to take the altitude of an object in degrees.

Gunter’s scale or Gunter’s rule, generally called the “Gunter” by seamen, is a large plane scale, usually 2 feet (0.61 m) long by about 1½ inches broad (600 mm by 40 mm), and engraved with various scales, or lines. On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, etc.), and on the other side the corresponding artificial or logarithmic ones. By means of this instrument questions in navigation, trigonometry, etc., are solved with the aid of a pair of compasses. It is a predecessor of the slide rule, a calculating aid used from the 17th century until the 1970s. Gunter’s line, or line of numbers refers to the logarithmically divided scale, like the most common scales used on slide rules for multiplication and division.

Gunter died at Gresham College on Dec 10, 1626, and was buried one day later in the churchyard of St Peter-le-Poer, Old Broad Street.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Edmund Gunter“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Edmund Gunter, English mathematician, at Britannica Online
- [3] “Gunter, Edmund.” Complete Dictionary of Scientific Biography. . Encyclopedia.com.
- [4] Edmund Gunter at Wikidata
- [5] Timeline for Edmund Gunter, via Wikidata

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]]>The post Paul Bernays and the Unified Theory of Mathematics appeared first on SciHi Blog.

]]>On October 17, 1888, Swiss mathematician and logician **Paul Isaac Bernay**s was born. Bernays made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. Bernays is known for his attempts to develop a unified theory of mathematics.

Bernays was born the oldest of five children in London, UK, the son of Julius Bernays, a businessman, and his mother was Sarah Brecher. He spent a happy childhood in Berlin, and attended the Köllner Gymnasium, 1895-1907. Bernays’s decision was to take up engineering and he entered the Technische Hoschule in Charlottenburg where he began his studies in 1907. However, despite his parents’ wish that he put his mathematical talents to practical use, Bernays decided after one semester (the 1907 summer semester) that he must make the change from engineering to pure mathematics.[1] At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. In 1910 he changed to the University of Göttingen, where he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. His very first publication actually was in philosophy, namely *Das Moralprinzip bei Sidgwick und bei Kant* (1910).[1]

In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Edmund Landau, on the analytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him the Habilitation for a thesis on complex analysis and Picard’s theorem. The examiner was Ernst Zermelo. Bernays became Privatdozent at the University of Zurich, 1912–17, where he came to know George Pólya.

At the University of Zürich, Bernays was appointed as a privatdocent and an assistant to Ernst Zermelo. He worked there until 1917 but this was not a productive period for him.[1] Starting in 1917, David Hilbert, on the occasion of his lecture in Zürich on “*Axiomatisches Denken*”, employed Bernays to assist him with his investigations of the foundations of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second Habilitation, for a thesis in which he established the completeness of propositional logic; this was in fact a study of Russell and Whitehead’s Principia Mathematica, and uses ideas from Ernst Schröder. This, however, was not published until 1926, and then only in abridged form.

In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. In 1933, he was dismissed from this post because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to Switzerland, whose nationality he had inherited from his father, and where the ETH employed him on occasion. He visited Princeton in session 1935-36 and gave courses on mathematical logic and set theory. In 1939 the ETH granted him the right to teach, but only for four years. It was extended in 1943 when the four year term was up. He obtained a half-time permanent post at the ETH from 1945 but there has been criticism of the ETH for not treating a distinguished academic like Bernays in a more honourable way. However Bernays never saw it that way and he was extremely grateful to the ETH for coming to his rescue at a time of great difficulty. He held this part-time post until 1959 when he retired and was made professor emeritus.[1]

Bernays’s collaboration with Hilbert culminated in the two volume work *Grundlagen der Mathematik* by Hilbert and Bernays (1934, 1939), discussed in Sieg and Ravaglia (2005). In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic, Bernays set out an axiomatic set theory whose starting point was a related theory John von Neumann had set out in the 1920s. Von Neumann’s theory took the notions of function and argument as primitive; Bernays recast von Neumann’s theory so that classes and sets were primitive. Bernays’s theory, with some modifications by Kurt Gödel, is now known as von Neumann–Bernays–Gödel set theory. A proof from the *Grundlagen der Mathematik* that a sufficiently strong consistent theory cannot contain its own reference functor is now known as the Hilbert–Bernays paradox. In 1956, Bernays revised Hilbert’s ‘*Grundlagen der Geometrie*’ (1899) on the foundations of geometry. He believed that the whole structure of mathematics could be combined as a single logical entity.

Bernays never married but after he moved to Zürich he lived with his mother and two unmarried sisters. After his mother died in 1953, he continued to live with his sisters. He remained research active well into his 80s. Paul Bernays passed away on 18 September 1977, aged 88.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Paul Bernays“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Paul Bernays: A Short Biography (1976)
- [3] Paul Bernays: Platonism in Mathematics, 1935
- [3] Paul Isaac Bernays at Mathematics Genealogy Project
- [4] Paul Bernays at zbMATH
- [5] Paul Isaac Bernays at Wikidata

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]]>The post Ivan Matveevich Vinogradov and the Goldbach Conjecture appeared first on SciHi Blog.

]]>On September 14, 1891, Soviet mathematician **Ivan Matveevich Vinogradov** was born. Vinogradov is best known for his contributions to the analytical theory of numbers, including a partial solution of the Goldbach conjecture proving that every sufficiently large odd integer can be expressed as the sum of three odd primes.

Ivan Matveevich Vinogradov was born to Matvei Avraam’evich Vinogradov, a priest in Milolyub, a village in the Velikie Luki district of the Pskov province of Russia. By 1903 Ivan’s father was a priest at the Church of the Holy Shroud in Velikie Luki, where Ivan attended school from 1903 to 1910. He entered the faculty of Mathematics and Physics at the University of St. Petersburg in 1910 under A. A. Markov and Ya. V. Uspenskii, who both had an interest in probability and number theory and should influence Vinogradov’s interest. He graduated B.A. in 1914 with a work on the distribution of quadratic residues and non-residues, and was awarded a masters degree in 1915.

Vinogradov was very single minded in his approach to mathematics and succeeded to press ahead with deep research despite the difficulties arising first from World War I. He taught at the State University of Perm, founded in 1916, originally as a branch of the University of St. Petersburg, from 1918 to 1920, and in 1920 he became a Professor at the St Petersburg Polytechnic Institute as well as docent at the St. Petersburg University. In 1925, he was promoted to professor at the university, becoming head of the probability and number theory section.

From 1934 on he was the first Director of the Steklov Mathematical Institute at the USSR Academy of Sciences in Leningrad, a position he held for the rest of his life, except for the five-year period (1941–1946) when the institute was directed by Academician Sergei Sobolev. In 1941 he was awarded the Stalin Prize. In 1951 he became a foreign member of the Polish Academy of Sciences and Letters in Kraków.

In analytic number theory, Vinogradov’s method refers to his main problem-solving technique, applied to central questions involving the estimation of exponential sums. The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916. In the 1920s the work of Hardy and Littlewood developed Weyl’s methods to attack other problems in analytic number theory. In its most basic form, it is used to estimate sums over prime numbers, or Weyl sums. It is a reduction from a complicated sum to a number of smaller sums which are then simplified.

With the help of this method, Vinogradov tackled questions such as the ternary Goldbach problem in 1937 (using Vinogradov’s theorem, published in Some theorems concerning the theory of prime numbers, 1937 ), and the zero-free region for the Riemann zeta function. His own use of it was inimitable; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure. He also used this technique on the Dirichlet divisor problem, allowing him to estimate the number of integer points under an arbitrary curve. This was an improvement on the work of Georgy Voronoy.

In 1918 Vinogradov proved the Pólya–Vinogradov inequality for character sums. Vinogradov received the highest honour the USSR Academy of Sciences could give, namely the Lomonosov Gold Medal. He was elected to the Royal Society of London in 1942 and to the London Mathematical Society in 1939.

Ivan Vinogradov died on 20 March 1983, aged 91.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Ivan Vinogradov“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Ivan Matveyevich Vinogradov, Soviet Mathematician, at Britannica Online
- [3] Scientific papers of Ivan M. Vinogradov at zbMATH
- [4] Ivan Matveevich Vinogradov at Wikidata, Timeline for Ivan M. Vinogradov via Wikidata

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]]>The post Carl Størmer and the Aurorae appeared first on SciHi Blog.

]]>On September 3, 1874, Norwegian mathematician and geophysicist Carl Størmer was born. Carl Størmer is known both for his work in number theory and for studying the movement of charged particles in the magnetosphere and the formation of aurorae. He also contributed both important photographic observations and mathematical data to the understanding of the polar aurora, of stratospheric and mesospheric clouds, and of the structure of the ionosphere. The discovery of the Van Allen Radiation Belts by James Van Allen confirmed with surprising accuracy Størmer’s theoretical analysis of solar charged particle trajectories in Earth‘s magnetic field.

Carl Størmer studied mathematics at the Royal Frederick University and later moved to Paris where he studied together with Picard, Poincaré, Painlevé, Jordan, Darboux, and Goursat at the Sorbonne. Størmer was appointed professor of mathematics at Kristiania in 1903 and he was elected the first president of the newly formed Norwegian Mathematical Society in 1918.

It is believed that when Størmer observed Kristian Birkeland’s experimental attempts to explain the aurora borealis, he was fascinated by aurorae and related phenomena. His first work on the subject attempted to model mathematically the paths taken by charged particles perturbed by the influence of a magnetized sphere, and Størmer eventually published over 48 papers on the motion of charged particles.

Størmer was able to show that the radius of curvature of any particle’s path is proportional to the square of its distance from the sphere’s center. To solve the resulting differential equations numerically, he used Verlet integration, which is therefore also known as Störmer’s method.

Størmer’s predicted particle motions were later verified by Ernst Brüche and Willard Harrison Bennett. Størmer’s calculations showed that small variations in the trajectories of particles approaching the earth would be magnified by the effects of the Earth’s magnetic field, explaining the convoluted shapes of aurorae.

Further, Størmer considered the possibility that particles might be trapped within the magnetic field, and worked out the orbits of these trapped particles, a prediction that was borne out after his death by the 1958 discovery of the Van Allen radiation belt. Størmer, a keen photographer, also took pictures of around 20 different observatories across Norway. By measuring their heights and latitudes by triangulation, Størmer discovered that the aurora are typically as high as 100 kilometers above ground. He managed to classify them by their shapes and discovered the “solar-illuminated aurora” where the upper parts of an aurora are lit by the sun. Størmer published his work in several book, including From the depths of space to the heart of the atom and The Polar Aurora.

**References and Further Reading**

- Carl Størmer at MacTutor History of Mathematics archive, University of St Andrews
- Carl Størmer Biography at the Biographical Memoirs of Fellows of the Royal Society

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]]>The post Gaspard de Prony and the Prony Brake appeared first on SciHi Blog.

]]>On July 22, 1755, French mathematician and hydraulic engineer **Gaspard Clair François Marie Riche de Prony** was born. De Prony is best known for his efforts in the mechanization of calculations as well as for his invention of the eponymous “brake” to measure the performance of machines and engines.

Gaspard de Prony’s family name was Riche, the de Prony title having been bought by his parents. De Prony was educated at the Benedictine College at Toissei in Doubs. Prony’s father, a prominent lawyer, wished his son to follow him into a legal career and had him trained in the classics. But after convincing his father and after spending more than a year studying mathematics, in 1776 Prony entered the École des Ponts et Chaussées, from which he graduated in 1779 as the top student and remained for a further year in Paris.[1,2]

In 1780 he became an engineer with the École des Ponts et Chaussés and after three years in a number of different regions of France he returned to the École des Ponts et Chaussés in Paris 1783. He won the admiration of its director Jean Rondolphe Perronet [3], who in 1783 had Prony brought from the provinces to Paris to assist him. Prony’s defense of Perronet’s bridge at Neuilly led to his first memoir, *On the thrust of arches* (1783). and to friendship with mathematician Caspar Monge [4], who personally initiated him in advanced analysis and descriptive geometry.

In 1785 de Prony visited England on a project to obtain an accurate measurement of the relative positions of the Greenwich Observatory and the Paris Observatory. In 1787 he was promoted to inspector at the École des Ponts et Chaussés. Further promotion in 1790 was followed the next year by his being appointed Engineer-in-Chief of the École des Ponts et Chaussés.[1] In 1791, de Prony embarked on the task of producing logarithmic and trigonometric tables for the French Cadastre. The effort was sanctioned by the French National Assembly, which, after the French Revolution wanted to bring uniformity to the multiple measurements and standards used throughout the nation. In particular, his tables were intended for precise land surveys, as part of a greater cadastre effort.

Inspired by Adam Smith‘s *Wealth of Nations*, de Prony divided up the labor into three levels, bragging that he “could manufacture logarithms as easily as one manufactures pins.”

- The first level consisted of five or six high-ranking mathematicians with sophisticated analytical skills, including Adrien-Marie Legendre and Lazare Carnot, who chose the analytical formulas most suited to evaluation by numerical methods, and specified the number of decimals and the numerical range the tables were to cover.
- The second group of lesser mathematicians combined analytical and computational skills, and this group calculated the pivotal values using the formulas provided and the sets of starting differences includingtemplates for the human computers, and the first worked row of calculations, as well as the instructions for the computers to carry the sequence to completion.
- The third group consisted of human “computers” (as they were called) , who had no more than a rudimentary knowledge of arithmetic and carried out the most laborious and repetitive part of the process.

Due to a lack of funding from inflation following the French revolution, the tables were never published in full.By the turn of the 19th century, there was a shift in the meaning of calculation. The talented mathematicians and other intellectuals who produced creative and abstract ideas were regarded separately from those who were able to perform tedious and repetitive computations. Before the 19th century, calculation was regarded as a task for the academics, while afterwards, calculations were associated with unskilled laborers. This was accompanied by a shift in gender roles as well, as women, who were usually underrepresented in mathematics at the time, were hired to perform extensive computations for the tables as well as other computational government projects until the end of World War II. This shift in the interpretation of calculation was largely due to de Prony’s calculation project during the French Revolution.

Prony saw this entire system as a collection of human computers working together as a whole – a machine governed by hierarchical principles of the division of labor. One of de Prony’s important scientific inventions was the “brake” which he invented in 1821 to measure the performance of machines and engines. Essentially the measurement is made by wrapping a cord or belt around the output shaft of the engine and measuring the force transferred to the belt through friction. The friction is increased by tightening the belt until the frequency of rotation of the shaft is reduced. In practice more engine power can then be applied until the limit of the engine is reached.

In 1798 de Prony refused Napoleon’s request that he join his army of invasion to Egypt. Fourier, Monge and Malus had agreed to be part of the expeditionary force and Napoleon was angry that de Prony would not come. After Napoleon was defeated the reorganisation in France included a reorganisation of the École Polytechnique which was closed during 1816. De Prony lost his position as professor there and was not part of the reorganisation committee. However, as soon as the school reopened, de Prony was asked to be an examiner so he continued his connection yet only had to work one month per year.[1]

Prony also was first to propose using a reversible pendulum to measure gravity, which was independently invented in 1817 by Henry Kater and became known as the Kater’s pendulum. He also created a method of converting sinusoidal and exponential curves into a systems of linear equations. Prony estimation is used extensively in signal processing and finite element modelling of non linear materials.

Prony was a member, and eventually president, of the French Academy of Science. He was also elected a foreign member of the Royal Swedish Academy of Sciences in 1810. His name is one of the 72 names inscribed on the Eiffel Tower. Gaspard de Prony died in 1839, aged 84.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Gaspard de Prony“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] “Prony, Gaspard-François-Clair-Marie Riche De.” Complete Dictionary of Scientific Biography. . Encyclopedia.com
- [3] Jean-Rondolphe Perronet and the Bridges of Paris, SciHi Blog, October 27, 2014.
- [4] Caspar Monge and the Geometry, SciHi Blog, May 10, 2014.

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