The post Eratosthenes and the Circumference of the Earth appeared first on SciHi Blog.

]]>**Eratosthenes of Cyrene** was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today. He is best known for being the first person to calculate the circumference of the Earth.

“Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.”

– Vitruvius, De Architectura, Book 1, Chap 6, Sec. 9

The son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Now part of modern-day Libya, Cyrene under Ptolemaic rule in the fourth and third century BC had become a place of cultivation, where knowledge blossomed. Like any young Greek, Eratosthenes would have studied in the local gymnasium, where he would have learned physical skills and social discourse as well as reading, writing, arithmetic, poetry, and music. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens.[1]

His interest in Plato led him to write his very first work at a scholarly level, *Platonikos*, inquiring into the mathematical foundation of Plato’s philosophies. Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus. He was a talented and imaginative poet. As a historian, Eratosthenes decided to work on giving a systematic chronography of the known world by figuring out the dates of literary and political events from the siege of Troy up until his time. This work was highly esteemed for its accuracy. George Syncellus was later able to preserve from *Chronographies* a list of 38 kings of the Egyptian Thebes. Eratosthenes also wrote *Olympic Victors*, a chronology of the winners of the Olympic Games.

The library at Alexandria was planned by Ptolemy I Soter (c. 367 BC – 283/2 BC) and the project came to fruition under his son Ptolemy II Philadelphus (309–246 BC). Ptolemy II Philadelphus appointed one of Eratosthenes’ teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion.

Eratosthenes made several important contributions to mathematics and science, and was a friend of Archimedes.[5] Around 255 BC, he invented the armillary sphere. An Armillary Sphere is a model of objects in the sky (in the celestial sphere), consisting of a spherical framework of rings, centered on Earth or the Sun, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic. In O*n the Circular Motions of the Celestial Bodies*, Cleomedes credited him with having calculated the Earth’s circumference around 240 BC, using knowledge of the angle of elevation of the Sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene (modern Aswan, Egypt).

Eratosthenes calculated the circumference of the Earth without leaving Egypt. He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He measured the Sun’s angle of elevation at noon on the same day in Alexandria. The method of measurement was to make a scale drawing of that triangle which included a right angle between a vertical rod and its shadow. This turned out to be 1/50th of a circle. Taking the Earth as spherical, and knowing both the distance and direction of Syene, he concluded that the Earth’s circumference was fifty times that distance. His knowledge of the size of Egypt was founded on the work of many generations of surveying trips. Pharaonic bookkeepers gave a distance between Syene and Alexandria of 5,000 stadia (a figure that was checked yearly). Some claim Eratosthenes used the Olympic stade of 176.4 m, which would imply a circumference of 44,100 km, an error of 10%.

Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the ‘Sieve of Eratosthenes‘ which, in modified form, is still an important tool in number theory research. The sieve appears in the Introduction to arithmetic by Nicomedes.[1] While Eratosthenes original work about his surprisingly accurate measurement is lost, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars.[1]

Eratosthenes also came up with a technique for charting the Earth’s surface. He separated the world known to him into a Northern and Southern division using an east–west line parallel to the equator running through the island of Rhodes and bisecting the Mediterranean. He added a second north–south line at right angles running through Alexandria. Eratosthenes drew additional east–west and north–south lines to his map, but instead of adding these lines in regular intervals, he drew them through famous places: Meroë (the capital of the ancient Ethiopian kings), the Pillars of Hercules, Sicily, the Euphrates River, the mouth of the Indus River and the tip of the Indian peninsula.[4]

Eratosthenes was afflicted by blindness in his old age, and he is said to have committed suicide by voluntary starvation.[2] Eratosthenes was the first antique scholar to call himself a “philologist”. By philology, he meant not only the study of linguistics and literature, but in a more general sense a multifaceted scholarship. Characteristic of his unbiased attitude toward deeply-rooted convictions is his criticism of poets, which did not spare even the highest authority like Homer. He did not approve of the truthfulness of the poets’ descriptions, since their goal was only entertainment and not instruction. Despite his fame and his extraordinary erudition, Eratosthenes did not become the founder of a school of his own. Of the four people named as his students in Suda, three cannot be identified with certainty and were therefore hardly important scientists. The fourth is the prominent grammarian Aristophanes of Byzantium, who succeeded Eratosthenes as head of the Library of Alexandria.

Carl Sagan – Cosmos – Eratosthenes, [9]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Eratosthenes“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Eratosthenes of Cyrene, Greek Scientist, at Britannica online
- [3] Courtnay Ast: Eratosthenes, at Wichita State University
- [4] Christian Violatti: Eratosthenes, at Ancient History Encyclopedia
- [5] Archimedes lifted the world off their Hinges, SciHi Blog
- [6] English translation of the primary source for Eratosthenes and the size of the Earth at Roger Pearse.
- [7] Rawlins, D. (2008). “Eratosthenes’s large Earth and tiny universe”.
*DIO*.**14**: 3–12 - [8] Eratosthenes at Wikidata
- [9] Carl Sagan – Cosmos – Eratosthenes, carlsagandotcom @ youtube
- [10] Dutka, J. (1993). “Eratosthenes’ measurement of the Earth reconsidered”.
*Arch. Hist. Exact Sci*.**46**(1): 55–66. - [11] Roller, Duane W. (2010).
*Eratosthenes’ Geography: Fragments collected and translated, with commentary and additional material*. Princeton: Princeton University Press. - [12] Smith, Sir William. “Eratosthenes”, in
*A Dictionary of Greek and Roman Biography and Mythology*. Ann Arbor, Michigan: University of Michigan Library, 2005. - [13] Timeline of Ancient Greek Mathematicians, via Wikidata and DBpedia

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]]>The post Maurice René Fréchet and the Theory of Abstract Spaces appeared first on SciHi Blog.

]]>On September 2, 1878, French mathematician **Maurice René Fréchet** was born. Fréchet is known chiefly for his contribution to real analysis. He is credited with being the founder of the theory of abstract spaces, which generalized the traditional mathematical definition of space as a locus for the comparison of figures; in Fréchet‘s terms, space is defined as a set of points and the set of relations. He also made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness.

Maurice Fréchet was born to a Protestant family in Maligny, Yonne, France, to Jacques Fréchet, a director of a Protestant orphanage in Maligny and later a head of a Protestant school, and his wife Zoé. After his father lost his job since the newly established Third Republic in France demanded all education to be secular, his mother set up a boarding house for foreigners in Paris. Maurice Fréchet attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard,[4] who recognised the potential of Fréchet and tutored him on an individual basis.

In 1900, he enrolled to École Normale Supérieure to study mathematics. He started publishing quite early, having published four papers in 1903. Seven further papers appeared in 1904, then remarkably eleven papers in 1905 as he undertook research for his doctorate under Hadamard’s supervision. There are different ways that people make major contributions to the progress of mathematics, some by solving the big questions, others by proposing new areas for research. Fréchet recognised himself that he fell into the latter category. In 1906 Fréchet wrote an outstanding doctoral dissertation *Sur quelques points du calcul fonctionnel*, where he started a whole new area with his investigations of functionals on a metric space and formulated the abstract notion of compactness, although he did not invent the name ‘metric space’ which is due to Hausdorff.[1,7] The resulting abstract spaces (such as metric spaces, topological spaces, and vector spaces) are characterized by their particular elements, axioms, and relationships. In particular, Fréchet devised a method of applying the notion of limits from calculus to the treatment of functions as elements of a vector space and a way of measuring lengths and distances among the functions to produce a metric space, which led to the profoundly fruitful subject now known as functional analysis.[2] The importance of the thesis is that it develops axiomatic analysis systems providing an abstraction of different objects studied by analysis in a similar way to group theory providing an abstraction of algebraic systems.[1]

From 1907–1908 he served as a professor of mathematics at the Lycée in Besançon, then moved in 1908 to the Lycée in Nantes to stay there for a year. After that he served at the University of Poitiers between 1910–1919. Fréchet had arranged to spend the academic year 1914-15 at the University of Illinois at Urbana in the United States and had accepted an appointment there for one year. He and his family were packed and ready to travel to the port to board their ship for the United States when World War I broke out and Fréchet was required for military service.[1]

At 1914 he was mobilised and because of his language skills was attached to the British Army as an interpreter. For the period of the war Fréchet retained his post at the Faculty of Science in Poitiers despite not being able to teach there. However before he was released from military service at the end of the war, he was selected to go to Strasbourg to assist with re-establishing the university there. He was both professor of higher analysis at the University of Strasbourg and Director of the Mathematics Institute there from 1919 to 1927.[1]

In 1928 Fréchet decided to move back to Paris, thanks to encouragement from Emile Borel,[5] who was then Chair in the Calculus of Probabilities and Mathematical Physics at the Sorbonne. Fréchet briefly held a position of lecturer at the Sorbonne’s Rockefeller Foundation and from 1928 was a Professor (without a Chair). Fréchet was promoted to tenured Chair of General Mathematics in 1933 and to Chair of Differential and Integral Calculus in 1935. In 1941 Fréchet succeeded Borel as Chair in the Calculus of Probabilities and Mathematical Physics, a position Fréchet held until he retired in 1949.

Fréchet’s book *Les espaces abstraits* was published in 1928. It is devoted almost exclusively to his work on general topology. Fréchet’s early influence as the pioneer of an effective theory of topology in abstract spaces was substantial, but in time his influence was superseded by that of Hausdorff, whose book became an important resource for students and scholars.[3] Although he published prolifically in the 1930s on probability and statistics, bringing functional analysis to bear, his contributions in these fields did not match in originality and importance his early work on topology and general analysis. Fréchet also introduced the concepts of uniform convergence and consistency. It was also Fréchet who was the first to use the term Banach space in 1928, whereby he called the *l ^{p}* sequence spaces Banach spaces at that time.

Despite his major achievements, Fréchet was not overly appreciated in France. As an illustration, while being nominated numerous times, he was not elected a member of the Academy of Sciences until the age of 78 in 1956. He was in correspondence with many important mathematicians, including the Russian mathematicians Nikolai Nikolayevich Lusin, Pavel Aleksandrov and Urysohn, with Frigyes Riesz, L. E. J. Brouwer, and with Polish mathematicians such as Waclaw Sierpinski and Kazimierz Kuratowski. He was a chevalier of the Légion d’Honneur, was elected to the Polish Academy of Sciences in 1929, and was an honorary member of the Royal Society of Edinburgh and a member of the International Institute of Statistics.

Arkady Etkin, *Topics In Analysis (Lecture 1) : Overview of Metric Spaces* [9]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Maurice René Fréchet“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Maurice Fréchet, French mathematician, at Britannica Online
- [3] “Fréchet, René Maurice.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. 1 Sep. 2016
- [4] Jacques Hadamard and the Description of Mathematical Thought, SciHi Blog
- [5] Émile Borel and the Infinite Monkey Problem, SciHi blog
- [6] Felix Hausdorff and the Basic Principles of Set Theory, SciHi blog
- [7] Maurice René Frèchet at zbMATH
- [8] Maurice René Frèchet at Mathematics Genealogy Project
- [9] Arkady Etkin,
*Topics In Analysis (Lecture 1) : Overview of Metric Spaces, Arkady Etkin @ youtube* - [10] “M.R. Fréchet (1878 – 1973)”. Royal Netherlands Academy of Arts and Sciences.
- [11] Frank Nielsen; Rajendra Bhatia (2012).
*Matrix Information Geometry*. Springer Science & Business Media. - [12] Maurice René Frèchet at Wikidata
- [13] Timeline for Maurice René Frèchet, via Wikidata

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]]>The post Carl Runge and the Early Days of Numerical Mathematics appeared first on SciHi Blog.

]]>On August 20, 1856, German mathematician, physicist, and spectroscopist **Carl Runge** (Carl David Tolmé Runge) was born. He was co-developer and co-eponym of the Runge–Kutta method , a single-step method for the approximate solution of initial value problems in numerical mathematics.

Carl Runge was born in Bremen, Germany, the son of the merchant Julius Runge and his wife Fanny Tolmé, who was from England. He spent his early childhood years in Havana, Cuba, where his father administered the Danish consulate. After Julius retired, the family returned to live permanently in Bremen, but Julius had only a short retirement for he died on 18 January 1864. Fanny was left on her own to bring up the eight children. In 1875 Carl Runge graduated from high school in Bremen, Germany. He then accompanied his now widowed mother to Italy for six months. He first studied literature and philosophy, then mathematics at the University of Munich. Runge attended courses with fellow student Max Planck and they became close friends, remaining so for the rest of their lives.[5]

In 1877 he continued his studies at the University of Berlin, where he was particularly influenced by the mathematicians Kronecker and Weierstrass. After receiving his doctorate in 1880 under Weierstrass and Kummer with a thesis in differential geometry entitled *Über die Krümmung, Torsion und geodätische Krümmung der auf einer Fläche gezogenen Curven (About the curvature, torsion and geodesic curvature of the curves drawn on a surface.)*, he habilitated in 1883.

After qualifying to be a Gymnasium teacher during session 1880-81, he completed the necessary examinations and returned to Berlin where he began to collaborate with Kronecker. In the spring of 1886, Runge became professor of mathematics at the Technical University of Hanover. In 1904, at the instigation of Felix Klein,[1] he was appointed to the newly created professorship of applied mathematics at the Georg August University of Göttingen, the first of its kind in Germany.

At first, his field of work was purely mathematical. From Kronecker he got the suggestion for number theory and from Weierstrass for function theory. During his time in Berlin, he learned about the Balmer series from his future father-in-law (in whose family he socialized a lot). Within a year of taking up the professorship at Hannover, Runge had moved away from pure mathematics to study the wavelengths of the spectral lines of elements other than hydrogen together with Heinrich Kayser. In Göttingen, together with Martin Wilhelm Kutta, he developed the Runge-Kutta method for the numerical solution of initial value problems. Also known is his investigation of interpolation polynomials and their behavior when the polynomial degree is increased (see Runge’s Phenomenon). In function theory he investigated the approximability of holomorphic functions and thus founded the Runge theory.

Runge undertook several major journeys. His knowledge of languages, especially English, was of great benefit to him. In 1897 he visited the meeting of the British Association in Toronto and subsequently all important American observatories. Together with Karl Schwarzschild he undertook a solar eclipse expedition to Algiers in 1906. In the winter of 1909, he went to Columbia University in New York for a year as an exchange professor. This was followed by a second tour of America, visiting not only universities and observatories but also the sites of his childhood in Havana. In 1923 he reached the age of retirement but he continued to lead his institute in Göttingen until in 1925 his successor Gustav Herglotz took over. In the summer of 1926, he attended the British Association meeting in Oxford.

Carl Runge died on January 3, 1927, of a heart attack, at age 70.

Runge Kutta Methods | Lecture 50 | Numerical Methods for Engineers, [13]

**References and Further Reading:**

- [1] Felix Klein and the Klein-Bottle, SciHi Blog
- [2] Karl Schwarzschild and the Event Horizon, SciHi Blog
- [3] Rudolf Fritsch:
*Runge, Carl David Tolmé.*In:*Neue Deutsche Biographie*(NDB). Band 22, Duncker & Humblot, Berlin 2005, ISBN 3-428-11203-2, S. 259 - [4] Paschen F (1929). “Carl Runge”.
*Astrophysical Journal*.**69**: 317–321. - [5] O’Connor, John J.; Robertson, Edmund F., “Carl David Tolmé Runge”,
*MacTutor History of Mathematics archive*, University of St Andrews - [6] Max Planck and the Quantum Theory, SciHi Blog
- [7] Carl Runge, Ueber die Krümmung, Torsion und geodätische Krümmung der auf einer Fläche gezogenen Curven (PhD dissertation, Friese, 1880)
- [8] Carl Runge, Graphical methods; a course of lectures delivered in Columbia university, New York, October, 1909, to January, 1910 (Columbia University Press, New York, 1912)
- [9] Iris Runge:
*Carl Runge und sein wissenschaftliches Werk*, Vandenhoeck & Ruprecht, Göttingen 1949. - [10] Carl Runge at the Mathematics Genealogy Project
- [11] Carl Runge at zbMATH
- [12] Carl Runge at Wikidata
- [13] Runge Kutta Methods | Lecture 50 | Numerical Methods for Engineers, Jeffrey Chasnov @ youtube
- [14] Timeline for Carl Runge, via Wikidata

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]]>The post Ferdinand Georg Frobenius and Group Theory appeared first on SciHi Blog.

]]>On August 3, 1917, German mathematician **Ferdinand Georg Frobenius** passed away. Frobenius best known for his contributions to the theory of elliptic functions, differential equations and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem.

“In mathematics, however, organizing talent plays a most subordinate role. Here weight is carried only by the individual. The slightest idea of a Riemann or a Weierstrass is worth more than all organisational endeavours. To be sure, such endeavours have pushed to take centre stage in recent years, but they are exclusively pursued by people who have nothing, or nothing more, to offer in scientific matters. There is no royal road to mathematics.”

– Ferdinand Georg Frobenius, [1]

Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, which by today is part of Berlin and actually rather close to the place where I am currently writing this article. His parents were Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. In 1860, he entered the Joachimsthal Gymnasium and after graduating in 1867, he went to the University of Göttingen, before returning to Berlin after only one semester. At the University of Berlin he attended lectures by Leopold Kronecker, Eduard Kummer and Karl Weierstrass.[4,5,6] Frobenius received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass with a thesis on the solution of differential equations.

In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics. Interestingly, Frobenius received a teaching position without completing a habilitation. Though today this is nothing extraordinary in Germany, by the time of Frobenius the German university system was much more formal and restrictive. It must have been made possible due to strong support by Weierstrass who was extremely influential and considered Frobenius one of his most gifted students.[1]

Frobenius was only in Berlin a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum, where he worked from 1875 to 1892. Influenced by Weierstrass, Frobenius was appointed to the vacant chair of Leopold Kronecker at University of Berlin in 1893, where he was elected to the Prussian Academy of Sciences. To justify his election, Weierstrass and Fuchs had listed 15 topics on which Frobenius already had made major contributions: the development of analytic functions in series, the algebraic solution of equations whose coefficients are rational functions of one variable, the theory of linear differential equations, Pfaff’s problem, linear forms with integer coefficients, linear substitutions and bilinear forms, adjoint linear differential operators, the theory of elliptic and Jacobi functions, the relations among the 28 double tangents to a plane of degree 4, Sylow’s theorem, double cosets arising from two finite groups, Jacobi’s covariants, Jacobi functions in three variables, the theory of biquadratic forms, and the theory of surfaces with a differential parameter.

In 1878 Frobenius proved Cayley-Hamilton ‘s theorem for matrices of any dimension. In 1877 he proved Frobenius’s theorem that there are only three associative finite-dimensional divisional algebras above the real numbers, the real numbers themselves, the complex numbers, and the quaternions.

As the major mathematics figure at Berlin, Frobenius continued the university’s antipathy to applied mathematics, which he thought belonged in technical schools. In some respects, this attitude contributed to the relative decline of Berlin in favour of Göttingen, which was less conservative in its views. On the other hand, he and his students made major contributions to the development of the modern concept of an abstract group — such emphasis on abstract mathematical structure became a central theme of mathematics during the 20th century.[2] Group theory was one of Frobenius’ principal interests in the second half of his career. One of his first contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today. Frobenius completely determined the characters of symmetric groups in 1900 and of characters of alternating groups in 1901, publishing definitive papers on each. He continued his applications of character theory in papers of 1900 and 1901 which studied the structure of Frobenius groups.[3]

Among the topics which Frobenius studied towards the end of his career were positive and non-negative matrices. He introduced the concept of irreducibility for matrices and the papers which he wrote containing this theory around 1910 remain today the fundamental results in the discipline. With Frobenius’s disdain for applied mathematics, it is somewhat ironic that his fundamental work in the theory of finite groups was later found to have surprising and important applications in quantum mechanics and theoretical physics.[2] Ferdinand Georg Frobenius died on 26. Oktober 1849 at age 67.

Javier Ribon *Herguedas, Lecture 56 Frobenius theorem*, [10]

** References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Ferdinand Georg Frobenius“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Georg Frobenius, German mathematician, at Britannica Online
- [3] Ferdinand Georg Frobenius, at Stetson University
- [4] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi blog, December 7, 2014.
- [5] Ernst Kummer and his Achievements in Mathematics, SciHi Blog, January 29, 2015.
- [6] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog
- [7] Ferdinand Georg Frobenius at Wikidata
- [8] Ferdinand Georg Frobenius at zbMATH
- [9] Ferdinand Georg Frobenius at Mathematics genealogy project
- [10] Javier Ribon
*Herguedas, Lecture 56 Frobenius theorem*, Javier Ribon Herguedas @ youtube - [11] Nikolaus Stuloff:
*Frobenius, Ferdinand Georg.*In:*Neue Deutsche Biographie*(NDB). Band 5, Duncker & Humblot, Berlin 1961 - [12] Timeline for Ferdinand Georg Frobenius, via Wikidata

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]]>The post The Astronomical Achievements of Sir George Biddell Airy appeared first on SciHi Blog.

]]>On July 27, 1801, English mathematician, astronomer, and Astronomer Royal **George Biddell Airy** was born. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as Astronomer Royal, establishing Greenwich as the location of the prime meridian.

“It is not simply that a clear understanding is acquired of the movements of the great bodies which we regard as the system of the world, but it is that we are introduced to a perception of laws governing the motion of all matter, from the finest particle of dust to the largest planet or sun, with a degree of uniformity and constancy, which otherwise we could hardly have conceived. Astronomy is pre-eminently the science of order.”

– George Biddell Airy, Popular Astronomy: A Series of Lectures Delivered at Ipswich (1868)

George Biddell Airy showed talents in mathematics very early at school and it is believed that he studied for instance arithmetic, double-entry book keeping and how to use a slide rule. He lived with his uncle for many years, who owned a large library with books on chemistry, biology, optics, as well as mechanics. In 1819, Airy entered Trinity College, Cambridge and graduated as Senior Wrangler in 1823. He was awarded a fellowship at Trinity and became determined to start a position with a decent financial status to be able to marry after unsuccessfully proposing to Richarda Smith, due to his inability to support her. George Biddell Airy was appointed Lucasian Professor of Mathematics at Cambridge only three years after he had graduated from Cambridge. However, his triumph in receiving the position was also followed by a longterm rivalry with Charles Babbage who also applied for the position. He only held the chair for just over a year, however, as he was appointed Professor of Astronomy and “Experimental Philosophy” in February 1828 and became head of the newly built Cambridge Observatory.

George Biddell Airy earned a good reputation as Plumian Professor of Astronomy at Cambridge and Director of the Cambridge Observatory. During his time at Cambridge Airy worked on mathematical, physical and astronomical problems. Among other things, he published papers on the refraction of lens glass (the so-called Airy discs are still used today to assess the quality of telescopes), the formation of rainbows and discovered the astigmatism of the human eye. He calculated the mass of the planet Jupiter and investigated the orbital disturbances of Earth and Venus. The latter work was very important and led to the improvement of astronomical tables. The Royal Astronomical Society awarded him its Gold Medal in 1833. Due to his increased salary, Airy was finally allowed to marry Richarda Smith in 1830. Five years later, Airy became Astronomer Royal and moved to Greenwich where he was for example responsible for the renovation of the observatory’s instruments. [1,2] In Airy’s opinion, the working conditions at the observatory were inadequate, so he reorganized the entire operation. He revised the existing records, secured the basis of a library, had an equatorial telescope installed by Richard Sheepshanks and an observatory for measuring the earth’s magnetic field.

During his academic career, Airy wrote the text ‘*On the Algebraic and Numerical Theory of Errors of Observations and the Combinations of Observations*‘. Even though it was supposed to be quite hard to read, it possibly influenced Pearson. Further notable works by Airy are ‘*Trigonometry*‘, ‘*Gravitation*‘, as well as ‘*Partial differential equations*‘. George Biddell Airy made major contributions to mathematics and astronomy and improved the orbital theory of Venus and the Moon. The scientist studied interference fringes in optics, and made a mathematical study of the rainbow. [1,2]

Another of Airy’s research objects was the determination of the mean density of planet Earth. He tackled the problem with the help of pendulums that he made swing on the surface and in the depths. In 1826, he began experiments at the Dolcoath Mine in Cornwall, but the pendulum, which was set up underground, was buried. Another experiment had to be abandoned due to a water ingress. Airy was only able to continue his experiments much later. Another experiment probably took place in 1854 at “Harton Pit”, near South Shields. Airy found out by means of the different oscillation frequencies that the gravitation is greater at a depth of 383 m than on the earth’s surface. For the density of the earth he derived a value of 6.566 g/cm3 (the actual value is 5.515 g/cm3), furthermore he derived a model of isostasia.

Next to his achievements in astronomy, George Biddell Airy was known to be interested in poetry as well as history, theology, architecture and many further subjects. In some fields, he even managed to publish papers including works on Julius Caesar. However, the opinions on Airy’s scientific achievements differ. Some point out Airy’s “failure” to discover planet Neptune. In the 1840s, Airy began to communicate with the French astronomer Urbain Le Verrier over the possibility that irregularities in the motion of Uranus were due to an unobserved body.[5] A few weeks later, he gave the order to systematically search for it in order to secure the triumph for Britain. Unfortunately for Airy however, Johann Gottfried Galle won the race and Airy basically took the blame.[6]

Still, due to his achievements, many see George Biddell Airy as one of the most important scientists of the 19th century. Through Airy’s activities the Royal Greenwich Observatory gained worldwide recognition. From 1872 to 1873 Airy was president of the Royal Astronomical Society. In 1881 he resigned from all official positions. He lived in the “White House” near the observatory until his death in 1892. [2,3]

Barton Zwiebach, *L8.1 Airy functions as integrals in the complex plane*, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “George Biddell Airy”,
*MacTutor History of Mathematics archive*, University of St Andrews. - [2] George Biddell Airy at Harvard
- [3] George Biddell Airy at Britannica
- [4] Charles Babbage – The Father of the Computer who hated Street Music, SciHi Blog
- [5] Urbain Le Verrier and the hypothetical Planet Vulcan, SciHi Blog
- [6] Johann Gottfried Galle and the First Observation of Planet Neptune, SciHi Blog
- [7] “Obituary – Sir George Biddell Airy”.
*Monthly Notices of the Royal Astronomical Society*.**52**: 212–229. 1892 - [8] Obituary in: .
*Popular Science Monthly*.**40**. April 1892. - [9] Works by or about George Biddell Airy at Internet Archive
- [10] George Biddell Airy at Wikidata
- [11] Barton Zwiebach,
*L8.1 Airy functions as integrals in the complex plane*, MIT OpenCourseWare @ youtube - [12] Timeline of English Astronomers, via Wikidata and DBpedia

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]]>The post Archytas – The Founder of Mathematical Mechanics appeared first on SciHi Blog.

]]>At about 428 BC, Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist **Archytas of Tarentum** was born. A scientist of the Pythagorean school he is famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.

“That tho’ a Man were admitted into Heaven to view the wonderful Fabrick of the World, and the Beauty of the stars, yet what would otherwise be Rapture and Extasie, would be but melancholy Amazement if he had not a Friend to communicate it to.”

— Attributed to Archytus by Christiaan Huygens, The Celestial Worlds Discover’d (1722)

Archytas was born in Tarentum, Magna Graecia, an area on the heel of the boot of Italy which was under Greek control in the 5th century BC, and was the son of Mnesagoras or Histiaeus. For a while, he was taught by Philolaus, a student of Pythagoras who, after Pythagoras died, fled to Lucania and then to Thebes in Greece. He returned to Italy where he taught Archytas. Archytas was a teacher of mathematics to Eudoxus of Cnidus [2]. As a Pythagorean, Archytas believed that only arithmetic, not geometry, could provide a basis for satisfactory proofs. Interestingly, Archytas is counted as pre-socratic philosopher, although there must have been an exchange of letters between Plato, i.e. Socrates’ student, and Archytas. Actually, Archytas sent a ship to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC.[3] This classification, however, is because of the style of Archytas’ philosophy rather than the strict chronology.[1]

Apart from the surviving fragments of his writings, our knowledge of Archytas’ life and work depends heavily on authors who wrote in the second half of the fourth century, in the fifty years after Archytas’ death. Archytas’ importance both as an intellectual and as a political leader is reflected in the number of writings about him in this period, although only fragments of these works have been preserved.[3]

Although Archytas studied many topics, since he was a Pythagorean, mathematics was his main subject and all other disciplines were seen as dependent on mathematics.[1] Archytas is believed to be the founder of mathematical mechanics. As only described in the writings of Aulus Gellius five centuries later, he was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters. This machine, which its inventor called *The pigeon*, may have been suspended on a wire or pivot for its flight. Another mechanical device invented by Archytas was a rattle for children which was useful, in Aristotle’s words

“… to give to children to occupy them, and so prevent them from breaking things about the house (for the young are incapable of keeping still).” (T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921))

Archytas also wrote some lost works, as he was included by Vitruvius in the list of the twelve authors of works of mechanics. Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not invent it. According to Eutocius, Archytas was the first to solve one of the most celebrated mathematical problems in antiquity, the duplication of the cube. He did it in his manner with the help of a geometric construction. Hippocrates of Chios before, reduced this problem to finding mean proportionals. He saw the ultimate goal of the sciences as the description of individual things in the world in terms of ratio and proportion and accordingly regarded logistic, the science of number and proportion, as the master science [3].

Archytas also developed a rather famous argument to show that the universe is unlimited in size: He asks anyone who argues that the universe is limited to engage in a thought experiment: “*If I arrived at the outermost edge of the heaven, could I extend my hand or staff into what is outside or not? It would be paradoxical [given our normal assumptions about the nature of space] not to be able to extend it.*” The end of the staff, once extended will mark a new limit. Archytas can advance to the new limit and ask the same question again, so that there will always be something, into which his staff can be extended, beyond the supposed limit, and hence that something is clearly unlimited.[3]

Archytas’ theory of proportions is treated in book VIII of Euclid’s Elements, where is the construction for two proportional means, equivalent to the extraction of the cube root. According to Diogenes Laertius, this demonstration, which uses lines generated by moving figures to construct the two proportionals between magnitudes, was the first in which geometry was studied with concepts of mechanics. The Archytas curve, which he used in his solution of the doubling the cube problem, is named after him. The Archytas curve is created by placing a semicircle (with a diameter of d) on the diameter of one of the two circles of a cylinder (which also has a diameter of d) such that the plane of the semicircle is at right angles to the plane of the circle and then rotating the semicircle about one of its ends in the plane of the cylinder’s diameter. This rotation will cut out a portion of the cylinder forming the **Archytas curve**.

Politically and militarily, Archytas appears to have been the dominant figure in Tarentum. The Tarentines elected him *strategos*, ‘general’, seven years in a row – a step that required them to violate their own rule against successive appointments. In his public career, Archytas had a reputation for virtue as well as efficacy. Some scholars have argued that Archytas may have served as one model for Plato’s philosopher king, and that he influenced Plato’s political philosophy as expressed in *The Republic* and other works.

Archytas’ Steam Powered Pigeon – Steam Culture, [8]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Archytas“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Eudoxus and the Method of Exhaustion, SciHi blog.
- [3] Huffman, Carl. “Archytas“. Stanford Encyclopedia of Philosophy.
- [4] Laërtius, Diogenes (1925). “Pythagoreans: Archytas“. Lives of the Eminent Philosophers 2:8.
- [5] Fragments of Archytas and Life of Archytas at Demonax, Hellenic Library
- [6] Archytas at Wikidata
- [7] Original works written by or about
*Archytas at Wikisource* - [8] Archytas’ Steam Powered Pigeon – Steam Culture, wareboilers @ youtube
- [9] von Fritz, Kurt (1970). “Archytas of Tarentum“.
*Dictionary of Scientific Biography*. Vol. 1. New York: Charles Scribner’s Sons. pp. 231–233. - [10] Timeline of Pythagoreans, via DBpedia and Wikidata

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]]>The post Roger Cotes and Newton’s Principia Mathematica appeared first on SciHi Blog.

]]>On July 10, 1682, English mathematician **Roger Cotes** was born. Cotes is well known for working closely with Isaac Newton by proofreading the second edition of his famous book, the *Philosophiae Naturalis Principia Mathematica*,[4] before publication. He also invented the quadrature formulas known as Newton–Cotes formulas and first introduced what is known today as Euler’s formula.

“If he had lived we would have known something.”, Remark of Issac Newton on the early death of Roger Cotes.

Cotes was born in Burbage, Leicestershire, England, as the second son to Reverend Robert Cotes, the rector of Burbage, and his wife Grace née Farmer. He attended Leicester School and by the age of twelve his teachers had already realized that he had an exceptional mathematical talent.[3] His aunt Hannah had married Reverend John Smith, and Smith took on the role of tutor to encourage Roger’s talent. The Smiths’ son, Robert Smith, would become a close associate of Roger Cotes throughout his life. Cotes later studied at St Paul’s School in London, still exchanging letters with his uncle on mathematical subjects, before he entered Trinity College, Cambridge in 1699 as a pensioner, meaning that he did not have a scholarship and paid for his own keep in College. He graduated BA in 1702, and was elected to a fellowship in 1705, even before he finished his MA in 1706.

In January 1706 he was nominated to be the first Plumian Professor of Astronomy and Experimental Philosophy. This was a remarkable achievement for Cotes who, at that time, was on 23 years of age. His extraordinary mathematical talent had been recognized by Isaac Newton as well as by Richard Bentley who was master of Trinity College. Both had recommended him for the chair, supported by William Whiston, Newton’s successor as Lucasian professor, despite the heavy opposition by John Flamsteed, astronomer royal,[5] who wanted his former assistant John Witty to be appointed.[3] It is not entirely clear how successful Cotes was in his role as an observational astronomer.[1]

Cotes began his educational career with a focus on astronomy. On his appointment to professor, he opened a subscription list in an effort to provide an observatory for Trinity. Unfortunately, the observatory still was unfinished when Cotes died, and was demolished in 1797.[2] In correspondence with Isaac Newton, Cotes designed a heliostat telescope with a mirror revolving by clockwork. He recomputed the solar and planetary tables of Giovanni Domenico Cassini [6] and John Flamsteed, and he intended to create tables of the moon’s motion, based on Newtonian principles. Cotes was elected a fellow of the Royal Society in 1711, and was ordained in 1713.

Cotes’ mathematical abilities put him second only to Newton from his generation in England.[1] From 1709 until 1713 much of Cotes’ time was taken up editing the second edition of Newton’s *Philosophiæ Naturalis Principia Mathematica*, originally published 1687 in the first edition only with a few printed books, where Newton described his principles of universal gravitation. In 1694 Newton did further work on his lunar and planetary theories, but illness and a dispute with Flamsteed postponed any further publication.[2] Cotes did not simply proof-read the work, rather he conscientiously studied it, gently but persistently arguing points with Newton.[1] The first edition of *Principia* had only a few copies printed and was in need of revision to include Newton’s works and principles of lunar and planetary theory. Newton at first had a casual approach to the revision, since he had all but given up scientific work. However, through the vigorous passion displayed by Cotes, Newton’s scientific hunger was once again reignited. The two spent nearly three and half years collaborating on the work, in which they fully deduce, from Newton’s laws of motion, the theory of the moon, the equinoxes, and the orbits of comets.

“But shall gravity be therefore called an occult cause, and thrown out of philosophy, because the cause of gravity is occult and not yet discovered? Those who affirm this, should be careful not to fall into an absurdity that may overturn the foundations of all philosophy. For causes usually proceed in a continued chain from those that are more compounded to those that are more simple; when we are arrived at the most simple cause we can go no father…. These most simple causes will you then call occult and reject them? Then you must reject those that immediately depend on them.” (from Roger Cotes’ Preface to Isaac Newton’s Principia)

Although Newton and Cotes seemed to have been at rather friendly terms in the beginning of their collaboration, his friendship seemed to have calmed down over time. In particular although Newton thanked Cotes in the first draft of a preface he wrote to this edition, he deleted these thanks for the final publication. Cotes formed a school of physical sciences at Trinity in collaboration with William Whiston. The two performed a series of experiments beginning in May 1707, the details of which can be found in a post-humous publication, *Hydrostatical and Pneumatical Lectures by Roger Cotes* (1738). [2]

Cotes’s major original work was in mathematics, especially in the fields of integral calculus, logarithms, and numerical analysis. He published only one scientific paper in his lifetime, titled *Logometria*, published in the *Philosophical Transactions of the Royal Society *for March 1714, in which he successfully constructs the logarithmic spiral. After his death, many of Cotes’s mathematical papers were hastily edited by his cousin Robert Smith, the successor on Cotes’ chair in Cambridge, and published in a book, *Harmonia mensurarum*, ensuring that he wasn’t forgotten, at least by the mathematical community.[3] Cotes’s additional works were later published in Thomas Simpson’s *The Doctrine and Application of Fluxions*. Although Cotes’s style was somewhat obscure, his systematic approach to integration and mathematical theory was highly regarded by his peers. Cotes discovered an important theorem on the nth roots of unity, foresaw the method of least squares, and he discovered a method for integrating rational fractions with binomial denominators. He was also praised for his efforts in numerical methods, especially in interpolation methods and his table construction techniques.

Cotes, who never married, died from a “Fever attended with a violent Diarrhoea and constant Delirium” in Cambridge in 1716 at the early age of 33. Isaac Newton remarked, “*If he had lived we would have known something*.”

Robin Wilson, *Pi and e and the most beautiful theorem in mathematics*, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Roger Cotes“, MacTutor History of Mathematics archive, University of St Andrews. (2005)
- [2] “Cotes, Roger.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com
- [3] Thony Christie, “If he had lived, we might have known something”, in The Renaissance Mathematicus, July 10, 2013.
- [4] Sir Isaac Newton and the famous Principia, SciHi Blog
- [5] John Flamsteed – Astronomer Royal, SciHi Blog
- [6] Giovanni Cassini and the Saturn Moon Rhea, SciHi Blog
- [7] Roger Cotes at Wikidata
- [8] Edleston, J. (ed.) (1850).
*Correspondence of Sir Isaac Newton and Professor Cotes*. via Internet Archive - [9] Roger Cotes at the Mathematics Genealogy Project
- [10] Roger Cotes at zbMATH
- [11] Robin Wilson,
*Pi and e and the most beautiful theorem in mathematics*, 2017, Gresham College @ youtube - [12] Timeline of Mathematical Analysts, via DBpedia and Wikidata

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]]>The post Jean-Victor Poncelet and Projective Geometry appeared first on SciHi Blog.

]]>On July 1, 1788, French engineer and mathematician **Jean-Victor Poncelet** was born, whose study of the pole and polar lines associated with conic led to the principle of duality. As a mathematician, his most notable work was in projective geometry. He developed the concept of parallel lines meeting at a point at infinity and defined the circular points at infinity that are on every circle of the plane. These discoveries led to the principle of duality, and the principle of continuity and also aided in the development of complex numbers.

“In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry… you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic … In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching.”

— Jean-Victor Poncelet

Jean-Victor Poncelet was born in Metz, France, on July 1, 1788, the illegitimate then legitimated son of Claude Poncelet, a lawyer and wealthy landowner, and Anne-Marie Perrein. At a young age, he was sent to live with the Olier family at Saint-Avold. At age fifteen, he returned to Metz for his secondary education, at Lycée Fabert followed by the École Polytechnique in Paris, from 1808 to 1810. After graduation, he joined the Corps of Military Engineers, attended the École d’Application, and achieved the rank of lieutenant in the French Army the same year he graduated.

Poncelet took part in Napoleon’s invasion of Russia in 1812, where he became a prisoner of war during Napoleon’s retreat from Moscow and was confined at Saratov. During his imprisonment he recalled the fundamental principles of geometry but, forgetting the details of what he had learnt from his teachers Caspard Monge, Carnot and Charles-Julien Brianchon, he went on to develop projective properties of conics.[2,5] He called the notes that he made the ‘*Saratov notebook* (*Cahiers de Saratov*),’ but it was only fifty years later that he incorporated much of what he had written in his treatise on analytic geometry *Applications d’analyse et de géométrie* (1862).[1] After the peace treaty was signed between France and Russia in 1814, Poncelet was released from Saratov and from 1815 he taught at Metz.

He published *Traité des propriétés projectives des figures* in 1822, which is a study of those properties which remain invariant under projection. This work was the first major to discuss projective geometry since Desargues‘, though Gaspard Monge had written a few minor works about it previously. It is considered the founding work of modern projective geometry. Joseph Diaz Gergonne also wrote about this branch of geometry at approximately the same time, beginning in 1810. Also his development of the pole and polar lines associated with conic sections led to the principle of duality (exchanging “dual” elements, such as points and lines, along with their corresponding statements, in a true theorem produces a true “dual statement”) and a dispute over priority with the German mathematician Julius Plücker for its discovery.[3]

From 1815 to 1825 Poncelet was a Captain of Engineers at Metz, overseeing the construction of machinery in the arsenal at Metz and teaching mechanics in the military college.[1] On 1 May 1820 he presented to the Académie des Sciences an important “*Essai sur les propriétés projectives des sections coniques*,” which contained the essence of the new ideas he wished to introduce into geometry. Poncelet sought to show — taking the example of the conics — that the language and concepts of geometry could be generalized by the systematic employment of elements at infinity and of imaginary elements. This goal was within reach, he contended, thanks to the introduction of the concept of “ideal chord” and the use of the method of central projections and of an extension procedure called the “principle of continuity.” But Augustin-Louis Cauchy,[6] the Academy’s referee, was little disposed to accept Poncelet’s high estimation of the value of geometric methods.[3] He claimed that the principle of continuity was “capable of leading to manifest errors”.[1]

In 1825, he became the professor of mechanics at the École d’Application in Metz, a position he held until 1835. During his tenure at this school, he improved the design of turbines and water wheels more than doubling the efficiency of the water wheel, deriving his work from the mechanics of the Provençal mill from southern France. It is hard for us to understand how important this work was for at this time much of industry was powered by waterwheels. Although the turbine of his design was not constructed until 1838, he envisioned such a design twelve years previous to that. Poncelet was promoted to Chef de Bataillon in 1831 and in 1835, he left École d’Application to become a tenured professor at Sorbonne (the University of Paris). He served on the Committee for Fortifications of Paris from 1835 to 1848.

In 1841 he became a Lieutenant-Colonel, then three years later became Colonel and, on 19 April 1848, a General of Brigade. He also became director of the École Polytechnique in April 1848, holding the post until 1850. During this time, he wrote *Applications d’analyse et de géométrie*, which served as an introduction to his earlier work *Traité des propriétés projectives des figures*. In 1849 Poncelet and Arthur Morin invented the dynamometer of rotation, which together with later refinements, became the basic investigative tool in the study of work. In 1851 the Great Exhibition of the Works of Industry of all Nations was held in Hyde Park, London. Poncelet was appointed as head of the Scientific Commission for the Exhibition. When the French organised the first Universal Exhibition in 1855. Poncelet also played an important role in this Exhibition.

After a long and painful illness, Jean-Victor Poncelet died in December 1867.

Poncelet’s theorem – a talk by Prof Joe Harris, [10]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Jean-Victor Poncelet“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Caspar Monge and the Geometry, SciHi blog, May 10, 2014.
- [3] Jean-Victor Poncelet, French mathematician, at Britannica Online
- [4] “Poncelet, Jean Victor.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
- [5] Nicolas Sadi Carnot and the Science of Thermodynamics, SciHi Blog
- [6] Augustin-Louis Cauchy and the Rigor of Analysis, SciHi blog
- [7] Jean-Victor Poncelet at zbMATH
- [8] Jean-Victor Poncelet at Wikidata
- [9] Chisholm, Hugh, ed. (1911). .
*Encyclopædia Britannica*.**22**(11th ed.). Cambridge University Press. p. 59. - [10] Poncelet’s theorem – a talk by Prof Joe Harris, Sicong Zhang @ youtube
- [11] Taton, René (1970). “Jean-Victor Poncelet”.
*Dictionary of Scientific Biography*. New York: Gale Cengage. - [12] Timeline for Jean-Victor Poncelet, via Wikidata

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]]>The post Henri Léon Lebesgue and the Theory of Integration appeared first on SciHi Blog.

]]>On June 28, 1875, French mathematician **Henri Léon Lebesgue** was born. He is best known for his theory of integration, which was a generalization of the 17th century concept of integration, i.e. summing the area between an axis and the curve of a function defined for that axis. By extending the work of Camille Jordan and Émile Borel on the Riemann integral, Lebesgue provided a generalization that solved many of the difficulties in using Riemann‘s theory of integration.

I heard of Lebesgue the first time in my basic analysis lectures during my first year at university. The funny thing I remember was that the analysis lab curses were taught by an older assistant with a rather heavy Saxonian accent. When we did the important proofs in integral calculus, he was always referring to Cauchy,[4] Lebesgue and Riemann,[5] and he had a very peculiar way to pronounce their names. Thus, when I read one of the names today, I immediately have his voice and his (funny) accent in my ears. But, let’s have a look at Lebesgue and his achievements.

“In my opinion a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy—an opinion, moreover, which has been expressed by many philosophers.

— Henri-Léon Lebesgue, as quoted in [13]

Henri Lebesgue was born in Beauvais, Oise, as son of a typesetter and a school teacher. His parents assembled at home a library that the young Henri was able to use. Unfortunately, Lebesgue‘s father died of tuberculosis when Henri was still very young and his mother had to support the family by herself. As he showed a remarkable talent for mathematics in primary school, one of his instructors arranged for community support to continue his education at the Collège de Beauvais and then at Lycée Saint-Louis and Lycée Louis-le-Grand in Paris.

Lebesgue entered the École Normale Supérieure in Paris in 1894 and was awarded his teaching diploma in mathematics in 1897. He was appointed professor at the Lycée Centrale at Nancy where he taught from 1899 to 1902. Building on the work of others, including that of Émile Borel [6] and Camille Jordan,[7] Lebesgue formulated the theory of measure in 1901 and in his famous paper *Sur une généralisation de l’intégrale définie* (1901), he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions – a work that revolutionised the integral calculus. Up to the end of the 19th century, mathematical analysis was limited to continuous functions, based largely on the Riemann method of integration.[1] Moreover, artificial restrictions were necessary to cope with discontinuities that cropped up with greater frequency as more exotic functions were encountered. The Riemann method of integration was applicable only to continuous and a few discontinuous functions. [2]

The first theory of integration was developed by Archimedes in the 3rd century BC with his method of exhaustion, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz [8] discovered the idea that integration was intrinsically linked to differentiation – a relationship which today is known the Fundamental Theorem of Calculus. However, mathematicians felt that Newton‘s and Leibniz‘s integral calculus did not have a rigorous foundation.[8] In the 19th century, Augustin Cauchy developed epsilon-delta limits, and Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral. To define this integral, one fills the area under the graph with smaller and smaller rectangles and takes the limit of the sums of the areas of the rectangles at each stage. For some functions, however, the total area of these rectangles does not approach a single number. As such, they have no Riemann integral.

The Lebesgue integral is a generalisation of the integral introduced by Riemann in 1854. In the folowing decades a measure-theoretic viewpoint was gradually introduced, most prominently by Camille Jordan’s treatment of the Riemann integral in his *Cours d’analyse* (1893). Placing integration theory within a measure-theoretic context made it possible for Lebesgue to see that a generalisation of the notions of measure and measurability carries with it corresponding generalisations of the notions of the integral and integrability. Building on the work of Émile Borel, Lebesgue completed Borel’s definitions of measure and measurability so that they represented generalisations of Jordan’s definitions and then used them to obtain his generalization of the Riemann integral.[3]

Lebesgue‘s idea was to first define measure, for both sets and functions on those sets. He then proceeded to build the integral for what he called simple functions; measurable functions that take only finitely many values. Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

In 1902 Lebesgue earned his Ph.D. from the Sorbonne with the seminal thesis on *Integral, Length, Area*, submitted with Borel as advisor. Subsequently, Lebesgue was offered a position at the University of Rennes, lecturing there until 1906, when he moved to the Faculty of Sciences of the University of Poitiers. This was in keeping with the standard French tradition of a young academic first having appointments in the provinces, then later gaining recognition in being appointed to a more junior post in the capital. In 1910 Lebesgue moved to the Sorbonne as a maître de conférences, being promoted to professor starting with 1919. In 1921 he left the Sorbonne to become professor of mathematics at the Collège de France, where he lectured and did research for the rest of his life.

In 1922 Lebesgue was elected a member of the Académie française. Henri Lebesgue died on 26 July 1941 in Paris.

Lecture 5. The Lebesgue integral and Monotone convergence theorem, [14]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Henri Léon Lebesgue“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Henri Lebesgue at Britannica Online
- [3] Lebesgue, Henri Léon. Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
- [4] Augustin-Louis Cauchy and the Rigor of Analysis, SciHi Blog
- [5] Bernhard Riemann’s novell approaches to Geometry, SciHi Blog
- [6] Émile Borel and the Infinite Monkey Problem, SciHi Blog
- [7] Camille Jordan and the Cours d’Analyse, SciHi Blog
- [8] Leibniz and the Integral Calculus, SciHi Blog
- [9] Henri Léon Lebesgue at Wikidata
- [10] Henry Léon Lebesgue at zbMATH
- [11] Henri Léon Lebesgue at Mathematics Genealogy Project
- [12] Burkill, J. C. (1944). “Henri Lebesgue. 1875-1941”.
*Obituary Notices of Fellows of the Royal Society*.**4**(13): 483–490. - [13] George Edward Martin,
*The Foundations of Geometry and the Non-Euclidean Plane*(1982), 19 - [14] Lecture 5. The Lebesgue integral and Monotone convergence theorem, Marcus Carlssons mathematics courses @ youtube
- [15] Timeline for Henri Léon Lebesgue, via Wikidata

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]]>The post Oswald Veblen and Foundations of modern Topology appeared first on SciHi Blog.

]]>On June 24, 1880, American mathematician, geometer and topologist **Oswald Veblen** was born. Veblen‘s work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905 while this was long considered the first rigorous proof, many now also consider Jordan‘s original proof rigorous.

“Mathematics is one of the essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry.”

– Oswald Veblen, in Bulletin of the American Mathematical Society, Volume 30 (1924), p. 289.

Oswald Veblen was born in Decorah, Iowa to Andrew Anderson Veblen, a teacher at Luther College in Decorah, Iowa, later professor of mathematics and physics at the University of Iowa, and his wife Kirsti. Veblen’s uncle was Thorstein Veblen, a well-known economist and social critic. Veblen went to school in Iowa City and did his undergraduate studies at the University of Iowa and Harvard University, where he was awarded a second B.A. in 1900. For his graduate studies, he went to study mathematics at the University of Chicago, where he obtained a Ph.D. in 1903. The head of the Chicago department was the highly regarded mathematician E. H. Moore. Among Moore’s interests was David Hilbert’s recent work on the foundations of geometry.[5] Veblen attended Moore’s fall 1901 seminar covering this topic. Hilbert had employed the undefined terms of point, line, and plane, to devise a scheme of 20 axioms. Questions soon arose over the independence of these axioms and Moore identified and sharpened the independence deficiencies in his seminar. Veblen was inspired to go further in his 1903 dissertation, *A System of Axioms for Geometry*. In it Veblen gave an axiom system based on point and order rather than on the traditional notions of point, line and plane. He presented twelve axioms for Euclidean geometry which he proved to be an complete system of axioms and he also proved the independence of the axioms.[1,4]

He continued in Chicago after the award of his doctorate as an associate in mathematics. Already at this time he had begun to undertake research in topology and he published *Theory on plane curves in non-metrical analysis situs* in 1905. Topology (from the Greek τόπος, place, and λόγος, study) in mathematics is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz,[6] who envisioned the *geometria situs* (Greek-Latin for “geometry of place”) and *analysis situs* (Greek-Latin for “picking apart of place”). Leonhard Euler’s Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field’s first theorems.[7]

Veblen taught mathematics at Princeton University from 1905 to 1932. During World War I, Veblen served first as a captain, later as a major in the army. He sought a commission in the Ordnance Department in the spring of 1917 and was appointed to the Office of Ballistics Research at the new Aberdeen Proving Ground in Maryland. In 1926, he was named Henry B. Fine Professor of Mathematics. In 1932, he helped organize the Institute for Advanced Study in Princeton, resigning his professorship to become the first professor at the Institute that same year. He had James Alexander, Albert Einstein, John von Neumann and Hermann Weyl, all mathematicians he had chosen, as original Institute members.[1] He kept his professorship at the Institute until he was made emeritus in 1950.

Veblen’s *Analysis Situs* (1922) was the first book to cover the basic ideas of topology systematically. It was his most influential work and for many years the best available topology text. Veblen also laid the foundations for topological research at Princeton.[2] Soon after the discovery of general relativity, Veblen turned to differential geometry and took a leading part in the development of generalized affine and projective geometry. His work *The Invariants of Quadratic Differential Forms* (1927) is distinguished by precise and systematic treatment of the basic properties of Riemannian geometry. In collaboration with his brilliant student John Henry Whitehead, Veblen extended the knowledge of the Riemann metric for more general cases in *The Foundations of Differential Geometry* (1932).

Veblen’s belief that “*the foundations of geometry must be studied both as a branch of physics and as a branch of mathematics*” quite naturally led him to the study of relativity and the search for a geometric structure to form a field theory unifying gravitation and electromagnetism. With respect to the Kaluza-Klein field theory, which involved field equations in five-dimensional space, he provided the first physical interpretation of the fifth coordinate. By regarding the coordinate as a gauge variable (a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations), Veblen was able to interpret the theory as one involving four-dimensional space-time. In connection with this contribution, Veblen provided a new treatment of spinors (expressions used to represent electron spin) that he summarized in *Projektive Relativitätstheorie* (1933; “*Projective Relativity Theory*”).

Veblen died in Brooklin, Maine, in 1960 at age 80. After his death the American Mathematical Society created an award in his name, called the Oswald Veblen Prize in Geometry. It is awarded every three years, and is the most prestigious award in recognition of outstanding research in geometry.

George Dyson, *Oswald Veblen And The Institute For Advanced Study*, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Oswald Veblen“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Oswald Veblen, American mathematician, at Britannica Online
- [3] “Veblen, Oswald.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
- [4] Steve Batterson: The Vision, Insight, and Influence of Oswald Veblen, Notices of the AMS, Volume 54, Number 5, pp. 606–619
- [5] David Hilbert’s 23 Problems, SciHi Blog
- [6] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [7] Read Euler, he is the Master of us all…, SciHi Blog
- [8] Oswald Veblen at Wikidata
- [9] Oswald Veblen at zbMATH
- [10] Oswald Veblen at the Mathematics Genealogy Project
- [11] George Dyson,
*Oswald Veblen And The Institute For Advanced Study*, Kurt Tazelaar @ youtube - [12] Works by or about Oswald Veblen at Internet Archive
- [13] Timeline for Oswald Veblen, via Wikidata

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