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

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

]]>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

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

]]>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

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

]]>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.

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

]]>The post Jordan Carson Mark and the Development of Thermonuclear Weapons appeared first on SciHi Blog.

]]>On July 6, 1913, Canadian mathematician **Jordan Carson Mark** was born. Mark is best known for his work on developing nuclear weapons for the United States at the Los Alamos National Laboratory. He oversaw the development of new weapons, including the hydrogen bomb in the 1950s. On the hydrogen bomb project he was able to bring together experts like Edward Teller, Stanislaw Ulam and Marshall Holloway despite their personal differences.

Jordan Carson Mark was born in Lindsay, Ontario, Canada. He received a Bachelor of Arts degree in mathematics and physics from the University of Western Ontario in 1935, and a Doctor of Philosophy (PhD) in mathematics from the University of Toronto in 1938 with his thesis “*On the Modular Representations of the Group GLH(3,P)*” under the supervision of Richard Brauer, a leading German American mathematician, who worked mainly in abstract algebra, but made important contributions to number theory.

Mark taught mathematics at the University of Manitoba, from 1938 until World War II, when he joined the Montreal Laboratory of the National Research Council of Canada in 1943. He came to the Los Alamos Laboratory in May 1945 as part of the British Mission to the Manhattan Project, although he remained a Canadian government employee. When World War II ended, Mark remained as part of the permanent staff of Los Alamos, becoming head of its Theoretical Division in 1947, a position previously held by Hans Bethe, which he remained in until he retired in 1973. In this position, Mark oversaw the development of various weapons systems, including thermonuclear bombs.

In 1947 the Los Alamos Laboratory, under the leadership of Norris Bradbury, who had succeeded Robert Oppenheimer, was much smaller than it had been during the war, because most of the wartime staff had returned to their universities and laboratories, but it was still the center of American nuclear weapons development, and the Theoretical Division was for many years the center of the laboratory. The Laboratory made great strides in improving the weapons, making them easier to manufacture, stockpile and handle. The Operation Sandstone tests in 1948 demonstrated that uranium-235 could be used in implosion-type nuclear weapons.

A crash program to develop the hydrogen bomb was approved by President Harry S. Truman in early 1950, after learning that the Soviets had tested a fission bomb, at Edward Teller‘s urging before the laboratory had a workable design. This put the laboratory under great pressure. When Stanislaw Ulam finally came up with a workable design, it was Mark that he approached first. Mark took the Ulam design to Bradbury, and they put the idea to Teller, who then completed and extended the invention. The Teller-Ulam design would become that of all thermonuclear weapons.[3,4]

When it came to testing the design in the Ivy Mike nuclear test, the test was successful, obliterating an island in Enewetak Atoll in the Pacific on November 1, 1952. As most weapon research in the 1960s no longer involved the Theoretical Division, Mark branched out, sponsoring research into hydrodynamics, neutron physics and transport theory. He also supported Frederick Reines’s research into neutrinos, for which Reines was awarded the Nobel Prize in Physics in 1995.[5]

In 1958, and again the following year, Mark was a scientific adviser to the United States delegation at the Conference of Experts on Detection of Nuclear Explosions in Geneva, where delegates from Western and Eastern bloc countries discussed detection methods in the context of negotiations that eventually led to the Partial Test Ban Treaty, which banned most forms of nuclear testing. He was committed to preventing nuclear weapons proliferation.

Mark was member of the American Mathematical Society and the American Physical Society. After he retired from Los Alamos in 1973 he served on the Nuclear Regulatory Commission’s Advisory Committee on Reactor Safeguards, and was a consultant for the Nuclear Control Institute.

J. Carson Mark died on March 2, 1997, aged 83.

**References and Further Reading:**

- [1] J. Carson Mark, at Atomic Heritage Foundation
- [2] T. Hilchey: J. Carson Mark, 83, Physicist In Hydrogen Bomb Work, Dies, The New York Times, March 9, 1997.
- [3] Please Don’t Ignite the Earth’s Atmosphere…, SciHi Blog, May 13, 2012.
- [4] Edward Teller and Stanley Kubrick’s Dr. Strangelove, SciHi Blog January 15, 2013.
- [5] Frederick Reines and the Neutrino, SciHi Blog, March 16, 2015.

The post Jordan Carson Mark and the Development of Thermonuclear Weapons appeared first on SciHi Blog.

]]>The post Prasanta Chandra Mahalanobis and Statistics appeared first on SciHi Blog.

]]>On June 29, 1893, Indian statistician Prasanta Chandra Mahalanobis was born. Mahalanobis is best remembered for the Mahalanobis distance, a statistical measure and for being one of the members of the first Planning commission of free India. He also made pioneering studies in anthropometry in India. He also devised fractile graphical analysis to study socioeconomic conditions. He applied statistics to issues of crop yields and planning for flood control.

Prasanta Chandra Mahalanobis earned his Bachelor of Science degree in 1912 and then left for England one year later in order to study at the University of London. Mahalanobis missed a train and stayed with a friend at King’s College, Cambridge. He was so impressed with the college that he decided to join it. After graduating, he was introduced to the journal Biometrika which he took with him to India. The journal sparked his enthusiasm for statistical problems in meteorology and anthropology.

In 1920, Mahalanobis met with the director of the Zoological Survey of India Nelson Annandale at a session of the Indian Science Congress led to Annandale asking him to analyse anthropometric measurements of Anglo-Indians in Calcutta. Mahalanobis intended to examine what factors influence the formation of European and Indian marriages. With the help of Annandale’s data collection and the caste specific measurements made by Herbert Risley, Mahalanobis found out that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. Further, he discovered that the intermixture more frequently involved the higher castes than the lower ones. Prasanta Chandra Mahalanobis explained his analysis in his paper published in 1922. While studying the topic, Mahalanobis found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted ‘D2’ and now eponymously named Mahalanobis distance, is independent of measurement scale.

At the Presidency College, Calcutta an informal group of scientists interested in statistics formed in the Statistical Laboratory. In 1931, Prasanta Chandra Mahalanobis, Nikhil Ranjan Sen, and Sir R. N. Mukherji established the Indian Statistical Institute, which was registered as a non-profit distributing learned society one year later. In 1959, the institute was declared as an institute of national importance and a deemed university.

Prasanta Chandra Mahalanobis is well known for his large-scale sample surveys. He introduced the concept of pilot surveys and advocated the usefulness of sampling methods. For instance, Mahalanobis worked on topics like consumer expenditure, tea-drinking habits, public opinion, crop acreage and plant disease. He further discovered a method for estimating crop yields which involved statisticians sampling in the fields by cutting crops in a circle of diameter 4 feet.

**References and Further Reading:**

- A Scientific Take on the Indian National Identity Through Mahalanobis’s Profiloscope at The Wire
- Pransanta Chandra Mahalanobis at Britannica Online
- Pransanta Chandra Mahalanobis Career and Major Works

The post Prasanta Chandra Mahalanobis and Statistics appeared first on SciHi Blog.

]]>The post Aryabhata and Indian Mathematics appeared first on SciHi Blog.

]]>In 476 CE, Indian mathematician and astronomer **Aryabhata** was born. Aryabhata is the earliest Indian mathematician whose work and history are available to modern scholars. In his work “Ganita” Aryabhata names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, using the decimal number system. He also came up with an approximation of pi and the area of a triangle.

“Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.” Bhāskara (c. 600 – c. 680), Indian mathematician

Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, adjacent to modern-day Patna, then the capital of the Gupta dynasty. It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Kusumapura became one of the two major mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.[2]

Aryabhata is the author of several treatises on mathematics and astronomy, most famously Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. Aryabhatiya was particularly popular in South India, where numerous mathematicians over the ensuing millennium wrote commentaries. The work was written in verse couplets. Following an introduction that contains astronomical tables and Aryabhata’s system of phonemic number notation in which numbers are represented by a consonant-vowel monosyllable, the work is divided into three sections: Ganita (“Mathematics”), Kala-kriya (“Time Calculations”), and Gola (“Sphere”).[1]

In Ganita Aryabhata names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, using the decimal number system. Then he treats geometric measurements—employing 62,832/20,000 (= 3.1416) for π—and develops properties of similar right-angled triangles and of two intersecting circles.[1] Other rules given by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians.[2]

With Kala-kriya Aryabhata turned to astronomy—in particular, treating planetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion, as well as planetary longitude corrections for different terrestrial locations. Aryabhatiya ends with spherical astronomy in Gola, where he applied plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes.[1] Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by two demons Rahu and Ketu. He explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth’s shadow. He discusses at length the size and extent of the Earth’s shadow and then provides the computation and the size of the eclipsed part during an eclipse.

Aryabhata’s work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata’s followers believed that the Earth rotated on its axis.

India’s first satellite Aryabhata launched in 1975 and the lunar crater Aryabhata are named in his honour.

**References and Further Reading:**

- [1] Aryabhata, Indian astronomer and mathematician, at Britannica Online
- [2] O’Connor, John J.; Robertson, Edmund F., “Aryabhata“, MacTutor History of Mathematics archive, University of St Andrews.
- [3] Aryabhata, at New World Encyclopaedia

The post Aryabhata and Indian Mathematics appeared first on SciHi Blog.

]]>The post Oliver Heaviside changed the Face of Telecommunications appeared first on SciHi Blog.

]]>On May 18, 1850, English self-taught electrical engineer, mathematician, and physicist Oliver Heaviside was born. Heaviside adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell’s field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

Oliver Heaviside suffered from scarlet fever as a child and had to deal with a hearing impairment since then. He left school at the age of 16, but continued to learn Morse code, and studied electricity. Heaviside’s uncle by marriage was Sir Charles Wheatstone, an expert in telegraphy and electromagnetism, and supporter of Heaviside’s further career. In 1867, he was sent to work with his older brother Arthur, who was managing one of Wheatstone’s telegraph companies in Newcastle-upon-Tyne.

At the Danish Great Northern Telegraph Company, Oliver Heaviside took a job as a telegraph operator and later became an electrician. Next to his work, Heaviside also kept studying and published an article in the prestigious Philosophical Magazine on ‘The Best Arrangement of Wheatstone’s Bridge for measuring a Given Resistance with a Given Galvanometer and Battery’. Further, Heaviside was able to show mathematically that uniformly distributed inductance in a telegraph line would diminish both attenuation and distortion. He further showed that if the inductance were great enough and the insulation resistance not too high, the circuit would be distortionless while currents of all frequencies would have equal speeds of propagation.

In 1880, Heaviside patented the coaxial cable in England and in 1884, he recast Maxwell’s mathematical analysis from its original cumbersome form to its modern vector terminology. Heaviside reduced twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell’s equations. The four re-formulated Maxwell’s equations describe the nature of electric charges, magnetic fields, and the relationship between the two, namely electromagnetic fields. In 1887, Oliver Heaviside collaborated on a paper titled “The Bridge System of Telephony” with his brother Arthur. However, Arthur’s superior William Henry Preece of the Post Office blocked the paper. It is believed that this development further fueled the long term animosity between Preece and Heaviside.

During the late 1880s, Heaviside became enthusiastic for electromagnetic radiation. He managed to calculate the deformations of electric and magnetic fields surrounding a moving charge, as well as the effects of it entering a denser medium. This achievement also included a prediction of what is now known as Cherenkov radiation, and inspired his friend George FitzGerald to suggest what now is known as the Lorentz–FitzGerald contraction. Heaviside proposed (what is now known as) the Kennelly–Heaviside layer of the ionosphere in 1902, which included means by which radio signals are transmitted around the Earth’s curvature.

**References and Further Reading:**

- Oliver Heaviside Biography by O’Connor and Robertson at MacTutor History of Mathematics Archive
- Oliver Heaviside at Britannica Online
- Oliver Heaviside at the Engineering and Technology Wiki

The post Oliver Heaviside changed the Face of Telecommunications appeared first on SciHi Blog.

]]>The post Pafnuty Chebyshev and the Chebyshev Inequality appeared first on SciHi Blog.

]]>On May 16, 1821, Russian mathematician Pafnuty Lvovich Chebyshev was born. Chebyshev is remembered primarily for his work on the theory of prime numbers, including the determination of the number of primes not exceeding a given number. Moreover, he is noted for his work in the fields of probability, statistics, mechanics, and number theory.

Pafnuty Chebyshev studied mathematical science at the University of Moscow starting from 1937. He later became Chebyshev became assistant professor of mathematics at the University of St. Petersburg. In 1874, Chebyshev became a foreign associate of the Institut de France.

Chebyshev is probably best known for developing an inequality of probability theory which was named Chebyshev’s inequality. It guarantees that, for a wide class of probability distributions, “nearly all” values are close to the mean. More exactly, no more than 1/k2 of the distribution’s values can be more than k standard deviations away from the mean. The inequality can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. Even though the theorem is named after Chebyshev, it was first formulated by his colleague Irénée-Jules Bienaymé who first stated the theorem without proof in 1853. Pafnuty Chebyshev later provided the proof and his student Andrey Markov provided another proof in his 1884 Ph.D. thesis. Further, the Bertrand–Chebyshev theorem states that for any n > 1, , there exists a prime number p such that n < p < 2 n. This is a consequence of the Chebyshev inequalities for the number π(n) of prime numbers less than n, which state that π(n) is of the order of n/log(n). A more precise form is given by the celebrated prime number theorem: the quotient of the two expressions approaches 1 as n tends to infinity.

Chebyshev is considered as one of the founding fathers of Russian mathematics. Among his well-known students were the mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has 10,629 mathematical “descendants” as of 2015

- Pafnuty Chebyshev at MacTutor History of Mathematics archive, University of St Andrews
- Pafnuty Chebychev at Britannica Online
- Chebyshev inequality in probability theory

The post Pafnuty Chebyshev and the Chebyshev Inequality appeared first on SciHi Blog.

]]>The post Alexis Clairaut and the Figure of the Earth appeared first on SciHi Blog.

]]>On May 13, 1713, French mathematician, astronomer, and geophysicist Alexis Claude Clairaut was born. Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton’s theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as “Clairaut’s theorem”. He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moon’s orbit.

Alexis Clairaut was born in Paris, France, to Jean-Babtiste Clairaut, who taught mathematics in Paris and was elected to the Brlin Academy, and his wife Catherine Petit. The couple had 20 children, however only a few of them survived childbirth. Jean-Baptiste Clairaut educated his son at home and set unbelievably high standards, using Euclid’s Elements while learning to read. However, Alexis was a child prodigy – by the age of nine he had mastered the excellent mathematics textbook of Guisnée, which provided a good introduction to the differential and integral calculus as well as analytical geometry. He went on to study de L’Hôpital’s books, in particular his famous text *Analyse des infiniment petits pour l’intelligence des lignes courbes*.

At the age of twelve Alexis wrote a memoir on four geometrical curves and under his father’s tutelage he made such rapid progress in the subject that in his thirteenth year he read before the Académie française an account of the properties of four curves which he had discovered. When only sixteen he finished a treatise on tortuous curves, *Recherches sur les courbes a double courbure*, which procured his admission into the French Academy of Sciences in 1729, but the king did not confirm his election until 1731. In July 1731 Clairaut became the youngest person ever elected to the Paris Academy of Sciences, at which time he was still only eighteen.[1]

In the Academy, Clairaut became interested in geodesy through Cassini’s work on the measurement of the meridian. He allied himself with Pierre Louis Maupertuis, who became a close friend, and the small but youthful and pugnacious group supporting Newton.[2] In 1736, together with Maupertuis, he took part in the expedition to Lapland, which was undertaken for the purpose of estimating a degree of the meridian arc. Maupertuis was director of the expedition, which included Le Monnier, Camus, the Abbé Outhier, and Celsius.[2] The goal of the excursion was to geometrically calculate the shape of the Earth, which Issac Newton theorized in his book *Principia* was an ellipsoid shape. They sought to prove if Newton’s theory and calculations were correct or not. Before the expedition team returned to Paris, Clairaut sent his calculations to the Royal Society of London.

Initially, Clairaut disagrees with Newton’s theory on the shape of the Earth. He outlined several key problems, as e.g. calculating gravitational attraction, the rotation of an ellipsoid on its axis, and the difference in density of an ellipsoid on its axes, that effectively disprove Newton’s calculations, and provided some solutions to the complications. He wrote:

“It appears even Sir Isaac Newton was of the opinion, that it was necessary the Earth should be more dense toward the center, in order to be so much the flatter at the poles: and that it followed from this greater flatness, that gravity increased so much the more from the equator towards the Pole.”[4]

This conclusion suggests not only that the Earth is of an oblate ellipsoid shape, but it is flattened more at the poles and is wider at the center. With his conclusion, he created much controversy, as he addressed the problems of Newton’s theory, but provided few solutions to how to fix the calculations. After his return, he published his treatise *Théorie de la figure de la terre* (1743), in which he promulgated the theorem, known as Clairaut’s theorem, which connects the gravity at points on the surface of a rotating ellipsoid with the compression and the centrifugal force at the equator. Under the assumption that the Earth was composed of concentric ellipsoidal shells of uniform density, Clairaut’s theorem could be applied to it, and allowed the ellipticity of the Earth to be calculated from surface measurements of gravity. This proved Sir Issac Newton’s theory that the shape of the Earth was an oblate ellipsoid. After his work on *Théorie de la figure de la Terre* Clairaut began to work on the three-body problem in 1745, in particular on the problem of the moon’s orbit. The first conclusions that he drew from his work was that Newton’s theory of gravity was incorrect and that the inverse square law did not hold. In this Clairaut had the support of Euler.[1]

Vivacious by nature, attractive, of average height but well built, Clairaut was successful in society and, it appears, with women.[2] However, he remaines unmarried. He was elected a Fellow of the Royal Society of London (1737) as well as the académies of Berlin, St. Petersburg, Bologna, and Uppsala. In 1743 Clairaut read before the Academy a Paper entitled “*L’orbite de la lune dans le systeme de M. Newton,*” Newton was not fully aware of the movement of the moon’s apogee, and therefore the problem had to be reexamined in greater detail. However, Clairaut—and d’Alembert, and Euler, who were also working on this question—found only half of the observed movement in their calculations. It was then that Clairaut suggested completing Newton’s law of attraction by adding a term inversely proportional to the fourth power of the distance.[2]

Clairaut guided the marquise du Châtelet in her studies, especially in her translation of Newton’s *Principia* and in preparing the accompanying explanations, a project which began before 1745 and continued until part of the book was published in 1756.[1,3,4] This translation of the *Principia* is elegant and, on the whole, very faithful to the original. It is, furthermore, most valuable because of its second volume, in which an abridged explanation of the system of the world is found. It illustrates, summarizes, and completes, on certain points, the results found in the *Principia*.[2] As a result of the experiments of Dortous de Mairan, Clairaut proved in 1735 that slight pendulum oscillations remain isochronous, even if they do not occur in the same plane.[2]

Clairaut died in Paris in 1765.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Alexis Clairaut“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] “Clairaut, Alexis-Claude.” Complete Dictionary of Scientific Biography. . Encyclopedia.com.
- [3] Sir Isaac Newton and the famous Principia, SciHi Blog, July 5, 2012.
- [4] A great man whose only fault was being a woman – Émilie du Châtelet, SciHi Blog,

December 17, 2013.

The post Alexis Clairaut and the Figure of the Earth appeared first on SciHi Blog.

]]>