The post Florence Nightingale – The Lady with the Lamp appeared first on SciHi Blog.

]]>On May 12, 1820, celebrated British social reformer and statistician **Florence Nightingale** was born. She is best known for being the founder of modern nursing. She came to prominence while serving as a nurse during the Crimean War, where she tended to wounded soldiers. She was known as “*The Lady with the Lamp*” after her habit of making rounds at night.

It is known that Florence Nightingale was a very well educated young woman and she realized the lack of opportunity for females in her social circle quite early. It is assumed that she had a very good relationship with her father, who was involved in anti-slavery movements. He was known to have respected his daughter as a friend and companion and supported her education highly. Florence started to visit people living in poverty and her interest in helping ill people increased highly in her early years. Florence often came to London to investigate possible occupations for women in the city’s hospitals. Unfortunately, nurses were not very much respected, since the occupation did not require a decent education at that time. Her visits became more frequent around 1844, but in this period, she also took time to travel to Egypt and Paris and she managed to get an introduction to a convent at Alexandria. She noticed that the disciplined and well-organised Sisters made better nurses than women in England. Following these events, Florence Nightingale attended the Institute of Protestant Deaconesses at Kaiserswerth, a training school for women teachers and nurses.

A major milestone in Florence Nightingale’s life was the beginning of the Crimean War in 1854. It was reported that sick and heavily wounded people suffered in English camps and in the media it was pointed out how different the conditions for wounded soldiers in French camps were. A shout out to all women in England was made to help the people of their country in need. Nightingale decided to go to the Crimea approximately in October of the same year and embarked with over thirty other nurses, reaching Scutari during the eve of the battle of Inkerman. Nightingale functioned as * Superintendent of the Female Nurses in the Hospitals in the East*, but everyone just called her the Lady in Chief. The headquarters for the newly arrived nurses was the barrack hospital at Scutari. The place was described as incredibly filthy and Nightingale explained that there was no water, soap, clothes or enough food when she arrived. The soldiers just laid there in their uniforms, spreading the infectious diseases. Next to the lack of sufficient supplies, the nurses also had to face the offensive behavior of the orderlies.

Fortunately, Nightingale and her crew managed to improve the situation at the hospital. More and more support, supplies and food were sent and Nightingale established a vast kitchen and a laundry. Next to the full time job at the hospital, Nightingale also took time to take care of the soldiers’ families and made rounds, watching the wounded soldiers at night since she was the only nurse allowed in the wards. The men started calling her the Lady with the Lamp. Henry Wadsworth Longfellow then popularized the phrase with his poem *Santa Filomena* (extract):

Lo! in that house of misery

A lady with a lamp I see

Pass through the glimmering gloom,

And flit from room to room.

With all the ill people in the hospital and the bad working condition for the doctors and nurses, many of them got sick or even passed away themselves. Also, the frost-bite and dysentery from exposure in the trenches before Sevastopol made the wards fuller than before and the death-rate increased to 42% by February 1855. Nightingale was not immune to the infectious diseases as well. When she visited Balaclava, she fell ill with the so called Crimean fever, but fortunately recovered and resumed her work in Scutari later on. In 1856, Florence Nightingale returned home and entered England without anyone noticing. She got the chance to meet Queen Victoria and Prince Albert, telling them about the miserable situation and then, a fund had been set up to found a training school for nurses.[5]

The Nightingale School and Home for Nurses was established at St. Thomas’s Hospital and she watched the progress of the new institution with practical interest even though she was asked to be its superintendent. Her health began to decline so she preferred to settle in London and retire from her busy work life. In this period, she published several works and reports on the army medical departments in the Crimea. In the following years, the first military hospital was established and an army medical college was opened at Chatham.

Nightingale immediately offered to leave for India when the Indian Mutiny broke out in 1857. Even though her services were not required, she became interested in the sanitary condition of the army and people there. From her work, a Sanitary Department was established in the Indian government. She became familiar with many facets of Indian life and demanded that there should be improvements in health and sanitation there.

Florence Nightingale exhibited a gift for mathematics from an early age and excelled in the subject under the tutelage of her father. Later, Nightingale became a pioneer in the visual presentation of information and statistical graphics. She used methods such as the pie chart, which had first been developed by William Playfair in 1801.[6] While taken for granted now, it was at the time a relatively novel method of presenting data. Indeed, Nightingale is described as “*a true pioneer in the graphical representation of statistics*“, and is credited with developing a form of the pie chart now known as the polar area diagram, or occasionally the Nightingale rose diagram, equivalent to a modern circular histogram, to illustrate seasonal sources of patient mortality in the military field hospital she managed. Nightingale called a compilation of such diagrams a “coxcomb”, but later that term would frequently be used for the individual diagrams. She made extensive use of coxcombs to present reports on the nature and magnitude of the conditions of medical care in the Crimean War to Members of Parliament and civil servants who would have been unlikely to read or understand traditional statistical reports. In 1859, Nightingale was elected the first female member of the Royal Statistical Society.

Florence Nightingale passed away on 13 August 1910.

Florence Nightingale and her Crimean War Statistics – Professor Lynn McDonald, [8]

**References and Further Reading:**

- [1] Florence Nightingale at the Victorian Web
- [2] Florence Nightingale at the BBC
- [3] Florence Nightingale at the Spartacus Educational Website
- [4] Website of the Florence Nightingale Museum
- [5] Victoria and Albert – A Royal Wedding, SciHi Blog
- [6] William Playfair and the Beginnings of Infographics, SciHi Blog
- [7] Florence Nighingale at Wikidata
- [8] Florence Nightingale and her Crimean War Statistics – Professor Lynn McDonald, Gresham College @ youtube
- [9] Timeline of British Reformers, via DBpedia and Wikidata

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]]>The post Diophantus of Alexandria – the father of Algebra appeared first on SciHi Blog.

]]>Probably sometime between AD 201 and 215, Alexandrian Greek mathematician **Diophantus of Alexandria** was born. He is often referred to as the father of algebra. He is the author of a series of books called *Arithmetica*, many of which are now lost, which deal with solving algebraic equations. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. The name “Diophantus” seems familiar? No wonder. While reading Claude Gaspard Bachet de Méziriac‘s edition of *Diophantus‘ Arithmetica*, famous mathematician Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found “*a truly marvelous proof of this proposition,*” now referred to as Fermat’s Last Theorem.[1]

“If we arrive at an equation containing on each side the same term but with different coefficients, we must take equals from equals until we get one term equal to another term. But, if there are on one or on both sides negative terms, the deficiencies must be added on both bides until all the terms on both sides are positive. Then we must take equals from equals until one term is left on each side.”

– Diophantus of Alexandria, as quoted in [8]

As for many other scientists of antiquity only little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 6th-century Greek anthology of number games and puzzles created by Metrodorus, a Greek grammarian and mathematician, who collected mathematical epigrams which appear in the *Greek Anthology*, a collection of poems that span the classical and Byzantine periods of Greek literature.

One of these epigrams is the famous epigram which reveals the age of Diophantus:

‘Here lies Diophantus,’ the wonder behold.

Through art algebraic, the stone tells how old:

‘God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life chill fate took him.

After consoling his fate by the science of numbers for four years, he ended his life.’

This can easily be transformed into the equation *x* = *x*/6 + *x*/12 + *x*/7 + 5 + *x*/2 + 4 that can be resolved to x=84. So he married at the age of 26 and had a son who died at the age of 42, four years before Diophantus himself died aged 84.[2] However, the age of 84 years for Diophantus nowhere else is confirmed.

The *Arithmetica* is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written.[2]

In the *Arithmetica* after some generalities about numbers, Diophantus first explains his symbolism: he uses symbols for the unknown (corresponding to our x) and its powers, positive or negative, as well as for some arithmetic operations—most of these symbols are clearly scribal abbreviations. This is the first and only occurrence of algebraic symbolism before the 15th century.[3] His style of algebra is known as the ‘syncopated’ style of algebraic writing, in which he represented polynomials as one unknown. Before Diophantus’s use of symbolism equations were written out completely.[4]

The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems. Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless. To give one specific example, he calls the equation *4 = 4x + 20* ‘absurd’ because it would lead to a meaningless answer. In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realized that a quadratic equation could have two solutions.[2]

In Book 3 of *Arithmetica*, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form *4n + 3* cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange [5] proved it using results due to Leonhard Euler.[6]

Diophantus himself refers to another work which consists of a collection of lemmas called *The Porisms* but this book is entirely lost. We do know three lemmas contained in *The Porisms* since Diophantus refers to them in the *Arithmetica*.[2] Diophantus is also known to have written on polygonal numbers, i.e. a number represented as dots or pebbles arranged in the shape of a regular polygon, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.

European mathematicians did not learn of the gems in Diophantus’s *Arithmetica* until Regiomontanus wrote in 1463:

“No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid…”[2], Regiomontanus, Oratio habita Patavii in praelectione Alfragani

Diophantus’ work has had a large influence in history. Editions of *Arithmetica* exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians.

Angela Stamilio, Diophantus, [10]

**Refenences and Further Reading:**

- [1] Pierre de Fermat and his Last Problem, SciHi blog.
- [2] O’Connor, John J.; Robertson, Edmund F., “Diophantus“, MacTutor History of Mathematics archive, University of St Andrews.
- [3] Diophantus of Alexandria, Greek Mathematician, in Britannica Online
- [4] Diophantus, at Famous Mathematicians
- [5] Joseph-Louis Lagrange and the Celestial Mechanics, SciHi Blog
- [6] Read Euler, he is the Master of us all…, SciHi Blog
- [7] Diophantus’s Riddle Diophantus’ epitaph, by E. Weisstein
- [8] Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History
- [9] Thomas Little Heath,
*Diophantos of Alexandria: A Study in the History of Greek Algebra*(1885) - [10] Angela Stamilio, Diophantus, MATH 455 Presentation 2 by Angela Stamilio @ youtube
- [11] Timeline of
**Ancient Greek Mathematicians**, via DBpedia and Wikidata

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]]>The post Stefan Banach and Modern Function Analysis appeared first on SciHi Blog.

]]>On March 30, 1892, Polish mathematician **Stefan Banach** was born. One of the founders of modern functional analysis, he is generally considered one of the world’s most important and influential 20th-century mathematicians. Some of the notable mathematical concepts that bear Banach‘s name include Banach spaces, Banach algebras, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach-Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.

“A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

– Stefan Banach, as quoted in [11]

Stefan Banach was born in Kraków, then part of the Austro-Hungarian Empire, to Stefan Grecze, a soldier in the Austro-Hungarian Army stationed in Kraków, and Katarzyna Banach. Since Stefan Greczek was only a private and thus prevented by military regulations from marrying, and the mother was too poor to support the child, the couple decided that the child should spent the first few years of his life with his grandmother, and later by Franciszka Płowa and her niece Maria Puchalska as foster family in Kraków. Banach was tutored by Juliusz Mien, a French intellectual and friend of the Płowa family, who most likely encouraged him in his early mathematical pursuits.

In 1902 Banach enrolled in Kraków’s Henryk Sienkiewicz Gymnasium, where he graduated in 1910. Banach together with his friend Witold Wiłkosz moved to Lwów to study at the Lwów Polytechnic. Both originally wanted to study mathematics, but they felt that nothing new could be discovered in mathematics. Thus, Banach chose to study engineering and Wilkosz chose oriental languages. It is almost certain that Banach, without any financial support from his father, had to support himself by tutoring. Thus, he graduated late in 1914 at age 22. When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision. When the Russian Army opened its offensive toward Lwów, Banach left for Kraków, where he spent the rest of the war. During the war he worked building roads but also spent time in Kraków where he earned money by teaching in the local schools. He also attended mathematics lectures at the Jagiellonian University in Kraków and, although this is not completely certain, it is believed that he attended famous Polish mathematician Stanisław Zaremba‘s lectures.[1]

In 1916, in Kraków’s Planty gardens, Banach encountered Hugo Steinhaus, one of the renowned mathematicians of the time. According to Steinhaus, while he was strolling through the gardens he was surprised to overhear the term “Lebesgue measure” and walked over to investigate. As a result, he met Banach discussing with his friends. Steinhaus became fascinated with the self-taught young mathematician and over the time developed a long-lasting collaboration and friendship. In fact, soon after the encounter Steinhaus invited Banach to solve some problems and soon the two published their first joint work (*On the Mean Convergence of Fourier Series*). Steinhaus, Banach and Nikodym, along with several other Kraków mathematicians also established a mathematical society, which eventually became the Polish Mathematical Society.

Banach was offered an assistantship to Antoni Łomnicki at Lvov Technical University in 1920. He lectured there in mathematics and submitted a dissertation for his doctorate under Lomnicki’s supervision. Steinhaus’ backing also allowed him to receive a doctorate without actually graduating from a university. The doctoral thesis *On Operations on Abstract Sets and their Application to Integral Equations* was published in 1922. It included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him in 1922 to become a professor at the Lwów Polytechnic. Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair.[1]

The years between the wars were extremely busy one for Banach. As well as continuing to produce a stream of important papers, he wrote arithmetic, geometry and algebra texts for high schools. He also was very much involved with the publication of mathematics. In 1929, together with Steinhaus, he started a new journal *Studia Mathematica* and Banach and Steinhaus became the first editors.[1] Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-metric space

In 1939, just before the start of World War II, Banach was elected as President of the Polish Mathematical Society. At the beginning of the war Soviet troops occupied Lwów. Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration. He was allowed to continue to hold his chair at the university and he became the Dean of the Faculty of Science at the university, now renamed the Ivan Franko University. Following the German takeover of Lwów in 1941, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl‘s Typhus Research Institute. Employment in Weigl’s Institute provided many unemployed university professors and their associates protection from random arrest and deportation to Nazi concentration camps.

After the Red Army recaptured Lwów in 1944, Banach returned to the University and helped re-establish it after the war years. However, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University. He was even considered a candidate for Minister of Education of Poland. In January 1945, however, he was diagnosed with lung cancer and died on August 31, 1945, aged 53.

Banach contributed to the theory of orthogonal series and made innovations in the theory of measure and integration, but his most important contribution was in functional analysis. Together with his coworkers Banach summarized the previously developed concepts and theorems of functional analysis and integrated them into a comprehensive system. His work started, of course, from what was achieved during the decades following Vito Volterra’s work of the 1890’s on integral equations [4]. Before Banach there were either rather specific individual results that only much later were obtained as applications of general theorems, or relatively vague general concepts. Ivar Fredholm’s and David Hilbert’s papers on integral equations marked the most substantial progress.[6] The concepts and theorems they had discovered later became an integral part of functional analysis, but most of them concern only a single linear space (later called Hilbert space).[3] Banach himself introduced the concept of normed linear spaces, which are now known as Banach spaces. He also proved several fundamental theorems in the field, and his applications of theory inspired much of the work in functional analysis for the next few decades.[2]

Claudio Landim, Doctorate program: Functional Analysis – Lecture 9: The Hahn-Banach theorem, [9]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Stefan Banach“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Stefan Banach, Polish Mathematician, at Britannica Online
- [3] “Banach, Stefan.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
- [4] Vito Volterra and Functional Analysis, yovisto blog, May 3, 2015.
- [5] Sheldon Axler: The Life of Stefan Banach, in American Mathematical Monthly 104 (1997), 577-579.
- [6] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [7] Stefan Banach at Wikidata
- [8] Stefan Banach at zbMATH
- [9] Doctorate program: Functional Analysis – Lecture 9: The Hahn-Banach theorem, Instituto de Matemática Pura e Aplicada @ youtube
- [10] Stefan Banach at the Mathematics Genealogy Project
- [11] Beata Randrianantoanina; Narcisse Randrianantoanina (2007). B
*anach Spaces and Their Applications in Analysis*: Proceedings of the International Conference at Miami University, May 22-27, 2006, in Honor of Nigel Kalton’s 60th Birthday. Walter de Gruyter. p. 5. ISBN 978-3-11-019449-4. - [12] Timeline for Stefan Banach, via Wikidata

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]]>The post Gösta Mittag-Leffler and the Acta Mathematica appeared first on SciHi Blog.

]]>On March 16, 1846, Swedish mathematician **Gösta Mittag-Leffler** was born. Mittag-Leffler‘s contributions are connected chiefly with the theory of functions. His mathematical research helped advance the Scandinavian school of mathematics. He is probably best known for founding the international mathematical journal Acta Mathematica.

“The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another. “

– Gösta Mittag-Löffler, quoted in [13]

Mittag-Leffler was born in Stockholm, as the eldest of four siblings to the school principal John Olof Leffler and Gustava Wilhelmina Mittag. He later added his mother’s maiden name to his paternal surname. Both sides of the family were of German origin but had lived for several generations in Sweden. Gösta also showed his many talents as he progressed through school, and his teachers in elementary school and later those at the Gymnasium in Stockholm realised that he had an outstanding ability for mathematics.[1] He matriculated at Uppsala University in 1865. During his studies he supported himself by taking private pupils. He completed his Ph.D. in 1872 with a thesis on applications of the argument principle and became docent at the university the same year.

Perhaps the event which would have the greatest lasting effect on his life was being awarded a salary which came through an endowment with a condition attached which said that the holder had to spend three years abroad. In 1873 Mittag-Leffler set off for Paris, where he attended Charles Hermite‘s lectures on elliptic functions.[12] From there, he continued to Göttingen and Berlin, where he studied under Weierstrass, which proved to be extremely influential in setting the direction of Mittag-Leffler’s subsequent mathematical work.[11]

He then took up a position as professor of mathematics at the University of Helsinki from 1877 to 1881 and then as the first professor of mathematics at the University College of Stockholm (the later Stockholm University). He was president of the college from 1891 to 1892 and retired from his chair in 1911. Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory. He worked on the general theory of functions, studying relationships between independent and dependent variables. His best known work concerned the analytic representation of a one-valued function, this work culminated in the Mittag-Leffler theorem. This study began as an attempt to generalise results in Weierstrass’s lectures where he had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity. Mittag-Leffler tried to generalise this result to meromorphic function while he was studying in Berlin.[1]

In 1882, Mittag-Leffler founded the mathematical journal *Acta Mathematica*, with the help of King Oscar’s sponsorship, and partly paid for with the fortune of his wife Signe Lindfors, who came from a very wealthy Finnish family. He also served as chief editor of the journal for 45 years. Ever since the start it has been one of the most prestigious mathematics journals in the world. The original idea for such a journal came from Sophus Lie in 1881, but it was Mittag-Leffler’s understanding of the European mathematical scene, together with his political skills, which ensured its success.[8] He attracted substantial contributions from the French mathematician Henri Poincaré,[9] and in the early volumes he demonstrated his support for Georg Cantor’s work in set theory by publishing French translations of Cantor’s papers.[3]

The journal’s “most famous episode” concerns Henri Poincaré, who won a prize offered in 1887 by Oscar II of Sweden for the best mathematical work concerning the stability of the Solar System by purporting to prove the stability of a special case of the three-body problem. The prize paper was to be published in *Acta Mathematica*, but after the issue containing the paper was printed, Poincaré found an error that invalidated his proof. He paid more than the prize money to destroy the print run and reprint the issue without his paper, and instead published a corrected paper a year later in the same journal that demonstrated that the system could be unstable. This paper later became one of the foundational works of chaos theory.

Mittag-Leffler went into business and became a successful businessman in his own right, but an economic collapse in Europe wiped out his fortune in 1922. He was a member of the Royal Swedish Academy of Sciences (1883), the Finnish Society of Sciences and Letters (1878, later honorary member), the Royal Swedish Society of Sciences in Uppsala, the Royal Physiographic Society in Lund (1906) and about 30 foreign learned societies, including the Royal Society of London (1896) and Académie des sciences in Paris. In 1883 Mittag-Leffler secured a position at the University of Stockholm for the Russian mathematician Sofya Kovalevskaya, the first woman to achieve such a post in modern Europe.[10] As a member of the Nobel Prize Committee in 1903, it was because of his intervention that the committee relented and awarded the prize for Physics to Marie Curie as well as her husband Pierre.

Gösta Mittag Leffler died in 1927, at age 81. He left a large estate of about 20,000 letters with 3000 correspondents (Mittag-Leffler also kept copies of sent letters), most of which is in the Royal Library in Stockholm. In the past it was occasionally claimed that Alfred Nobel did not endow a Nobel Prize in mathematics because he feared out of personal animosity that Mittag-Leffler would then be forced to receive a prize as a leading Swedish mathematician.

The Poincaré Conjecture (special lecture) John W. Morgan [ICM 2006], [14]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Gösta Mittag-Leffler“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Acta Informatica, original website
- [3] Magnus Gösta Mittag-Leffler, Swedish Mathematician, at Britannica Online
- [4] Henri Poincaré – the Last Universalist of Mathematics, SciHi Blog, April 29, 2014
- [5] Gösta Mittag-Leffler at zbMATH
- [6] Gösta Magnus Mittag-Leffler at Mathematics Genealogy Project
- [7] Gösta Mittag-Leffler at Wikidata
- [8] Sophus Lie and the Lie Theory, SciHi Blog, December 17, 2017.
- [9] Henri Poincaré – the Last Universalist of Mathematics, SciHi Blog, April 29, 2014.
- [10] Sofia Kovaleveskaya – Mathematician and Writer, SciHi Blog, January 15, 2016.
- [11] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog
- [12] Charles Hermite’s admiration for simple beauty in Mathematics, SciHi Blog
- [13] N. Rose, Mathematical Maxims and Minims (Raleigh N C 1988).
- [14] The Poincaré Conjecture (special lecture) John W. Morgan [ICM 2006], @ Graduate Mathematics @ youtube
- [15] Gilman, D. C.; Peck, H. T.; Colby, F. M., eds. (1905). .
*New International Encyclopedia*(1st ed.). New York: Dodd, Mead. - [16] Timeline of Swedish mathematicians, via DBpedia and Wikidata

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]]>The post Emil Artin and Algebraic Number Theory appeared first on SciHi Blog.

]]>On March 3, 1898, Austrian mathematician **Emil Artin** was born. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields.

Emil Artin was born in Vienna to parents Emma Maria Artin, a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, an opera singer and art dealer. He was brought up in the town of Reichenberg in Bohemia which was then part of the Austrian Empire.[1] Artin entered school in September 1904, presumably in Vienna. By then, his father was already suffering symptoms of advanced syphilis and died in 1906, when Artin was eight. In 1907, Artin entered the Volksschule in Strobnitz, a small town in southern Czechoslovakia near the Austrian border, followed by the Realschule in Reichenberg, where he pursued his secondary education until 1916. Rather surprisingly up to sixteen he did not show any particular talent for mathematics.

In October, 1916, Artin matriculated at the University of Vienna, having focused by now on mathematics. After one semester, however, he was drafted into the Austrian army and he served with this army until the end of the War. Then, in 1919, he entered the University of Leipzig where he continued his mathematics studies with Gustav Herglotz. Academic success came quickly and in 1921 he was awarded his doctorate. His thesis concerned applying the methods of the theory of quadratic number fields to quadratic extensions of a field of rational functions of one variable taken over a finite prime field of constants.[1]

In 1922, Artin went to the University of Hamburg as an assistant for the start of winter semester of session 1922-23. In 1923 he had his Habilitation and accordingly became Privatdozent at Hamburg. At Hamburg Artin lectured on a wide variety of topics including mathematics, mechanics and relativity. He was promoted to extraordinary professor there in 1925, then he became an ordinary professor in the following year being one of the youngest professors of mathematics in Germany at the age of 28.

These were particularly productive years for Artin’s research. He made a major contribution to field theory, the theory of braids and, around 1928, he worked on rings with the minimum condition on right ideals, now called Artinian rings. He had the distinction of solving, in 1927, one of the 23 famous problems posed by Hilbert in 1900 [4], in particular Hilbert’s seventeenth problem, which concerns the expression of positive definite rational functions as sums of quotients of squares.

Also in 1927 he gave a general law of reciprocity which included all previously known laws of reciprocity which had been discovered from the time that Gauss produced his first law.[1,5] Another important piece of work done by Artin during his first period in Hamburg was the theory of braids which he presented in 1925. He again showed his originality by introducing this new area of research which today is being studied by an increasing number of mathematicians working in group theory, semigroup theory, and topology.[1]

In 1929, Artin married one of his students, Natalie Jasny, called Natascha. One of their shared interests was photography, and when Artin bought a Leica A camera for their joint use, Natascha began chronicling the life of the family, as well as the city of Hamburg. For the next decade, she made a series of artful and expressive portraits of Artin that remain by far the best images of him taken at any age. Artin, in turn, took many fine and evocative portraits of Natascha.

In Fall 1937 Artin emigrated with his wife and family to the United States of America where he was teaching for a year at Notre Dame University, thereafter from 1938 until 1946 at Indiana University, Bloomington, and finally from 1946 until 1958 at Princeton University. Since Fall 1958, he was teaching again at Hamburg University where his life suddenly came to an end while he was still active.[2]

In 1944 he discovered rings with minimum conditions for right ideals, now known as Artin rings. He presented a new foundation for and extended the arithmetic of semi-simple algebras over the rational number field.[3] During his years in the United States Artin put his energies into teaching and supervising his Ph.D. students who themselves went on to make major contributions. He published relatively few papers, but he wrote a number of extremely important texts which have become classics.[1]

Artin’s books include *Geometric Algebra* (1957) and, with John T. Tate, *Class Field Theory* (1961).[3] Artin was one of the leading algebraists of the century, with an influence larger than might be guessed from the one volume of his *Collected Papers* edited by Serge Lang and John Tate. He worked in algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. The influential treatment of abstract algebra by van der Waerden is said to derive in part from Artin’s ideas, as well as those of Emmy Noether.[6] Artin also left two conjectures, both known as Artin’s conjecture. The first concerns Artin L-functions for a linear representation of a Galois group; and the second the frequency with which a given integer a is a primitive root modulo primes p, when a is fixed and p varies. These are unproven; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.

Galois theory I | Math History | NJ Wildberger, [10]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Emil Artin“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] H. Zassenhaus. Emil Artin, his life and his work. Notre Dame Journal of Formal Logic, 5:1–9, 1964.
- [3] Emil Artin, German mathematician, at Britannica Online
- [4] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [5] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog, April 30, 2013.
- [6] Emmy Noether and the Love for Mathematics, SciHi Blog, March 23, 2013.
- [7] Emil Artin at zbMATH
- [8] Emil Artin at Mathematics Genealogy Project
- [9] Emil Artin at Wikidata
- [10] Galois theory I | Math History | NJ Wildberger, Insights into Mathematics @ youtube
- [11] Albert, A. A. (1945). “Review of
*Galois theory*by Emil Artin with a chapter on applications by A. N. Milgram”.*Bull. Amer. Math. Soc*.**51**: 359 - [12] Schafer, Alice T. (1958). “Review of
*Geometric algebra*by E. Artin”.*Bull. Amer. Math. Soc*.**64**: 35–37 - [13] Timeline for Emil Artin, via Wikidata

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]]>The post Francois Viète and the Foundations of Algebra appeared first on SciHi Blog.

]]>On February 23, 1603, French mathematician **François Viète** passed away. Viète‘s work on a algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV.

Vieta was born at Fontenay-le-Comte, in western France about 50 km east of the coastal town of La Rochelle, to his father Etienne Viète, an attorney and a notary. He went to a Franciscan school and in 1558 studied law at the University of Poitiers, graduating as a Bachelor of Law in 1559. He began his career as an attorney in his native town. From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots. In 1564, Vieta entered the service of Antoinette d’Aubeterre, Lady Soubise, wife of Jean V de Parthenay-Soubise, one of the main Huguenot military leaders. He was employed to supervise the education of Soubise’s twelve year old daughter Catherine de Parthenay. He taught her science and mathematics and wrote for her numerous treatise on astronomy, geography and trigonometry, some of which have survived.

This was a period of great political and religious unrest in France. Charles IX had become king of France in 1560 and shortly after, in 1562, the French Wars of Religion began. It is a gross over-simplification to say that these wars were between Protestants and Roman Catholics but fighting between the various factions would continue on and off until almost the end of the century. [1]

In 1568, Antoinette, Lady Soubise, married her daughter Catherine to Baron Charles de Quellenec and Vieta went with Lady Soubise to La Rochelle. In 1571, Viète enrolled as an attorney in Paris, and continued to visit his student Catherine. He began publishing his *Universalium inspectionum ad canonem mathematicum liber singularis* and wrote new mathematical research by night or during periods of leisure.

“These things which are new are wont in the beginning to be set forth rudely and formlessly and must then be polished and perfected in succeeding centuries. Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms…”

(Francois Viete, Introduction to In Artem Analyticam Isagoge, 1591))

In 1572, Vieta was in Paris during the St. Bartholomew’s Day massacre on August 23. This must have been an extremely difficult time for Viète for, although not active in the Protestant cause, he was a Huguenot himself. That night, Baron De Quellenec was killed after having tried to save Admiral Coligny the previous night. The same year, Vieta met Françoise de Rohan, Lady of Garnache, and became her adviser against Jacques, Duke of Nemours.

In 1573, he became a councillor of the Parliament of Brittany, at Rennes, and two years later, he obtained the agreement of Antoinette d’Aubeterre for the marriage of Catherine of Parthenay to Duke René de Rohan, Françoise’s brother. In 1576, Henri, Duc de Rohan took him under his special protection, recommending him in 1580 as “maître des requêtes“. In 1579, Vieta printed his *canonem mathematicum*. In this tense atmosphere Viète was appointed by Henry III as royal privy counsellor on 25 March 1580, and he was attached to the parliament in Paris.[1]

Between 1583 and 1585, the League persuaded Henry III to release Vieta, Vieta having been accused of sympathy with the Protestant cause. Vieta retired to Fontenay and Beauvoir-sur-Mer, with François de Rohan. He spent four years devoted to mathematics, writing his “*Analytical Art*” or *New Algebra*. In 1589, Henry III took refuge in Blois. He commanded the royal officials to be at Tours before 15 April 1589. Vieta was one of the first who came back to Tours. He deciphered the secret letters of the Catholic League and other enemies of the king. Later, he had arguments with the classical scholar Joseph Juste Scaliger. Vieta triumphed against him in 1590.[5]

After the death of Henry III, Vieta became a Privy Councillor to Henry of Navarre, now Henry IV. He was appreciated by the king, who admired his mathematical talents. Vieta was given the position of councillor of the parlament at Tours. In 1590, Vieta discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.

In 1593, Vieta published his arguments against Scaliger [5]. Three years later, Scaliger resumed his attacks from the University of Leyden. Vieta replied definitively the following year. That same year, Belgian mathematician Adriaan van Roomen sought the resolution, by any of Europe’s top mathematicians, to a polynomial equation of degree 45. King Henri IV received a snub from the Dutch ambassador, who claimed that there was no mathematician in France. He said it was simply because some Dutch mathematician, Adriaan van Roomen, had not asked any Frenchman to solve his problem. Vieta came, saw the problem, and, after leaning on a window for a few minutes, solved it. It was the equation between sin(x) and sin(x/45). He resolved this at once, and said he was able to give at the same time (actually the next day) the solution to the other 22 problems to the ambassador. “*Ut legit, ut solvit*,” he later said.

In 1598, Vieta was granted special leave. Henry IV, however, charged him to end the revolt of the Notaries, whom the King had ordered to pay back their fees. Sick and exhausted by work, he left the King’s service in December 1602 and died soon after in 1603.

Vieta gave algebra a foundation as strong as in geometry. He then ended the algebra of procedures (*al-Jabr and Muqabala*), creating the first symbolic algebra. In doing so, he did not hesitate to say that with this new algebra, all problems could be solved (*nullum non problema solvere*). In his treatise *In artem analyticam isagoge* (Tours, 1591), Viète demonstrated the value of symbols introducing letters to represent unknowns. He suggested using letters as symbols for quantities, both known and unknown. He used vowels for the unknowns and consonants for known quantities. The convention where letters near the beginning of the alphabet represent known quantities while letters near the end represent unknown quantities was introduced later by Descartes in *La Gèometrie*.[6] This convention is used today, often without people realising that a convention is being used at all.[1] Thus he also used the signs + and -, first used by Johannes Widmann in a book in 1489, in his works. Previously, these had usually been written out as *plus* and *minus* in arithmetic operations. He also used the fraction bar as a symbol of division and the little word “in” as a fixed abbreviation of multiplication. Furthermore, Viète expressed the equality of two terms by the word “*aequabitur*” and thus invented the first equals sign. He also wrote related terms under each other and connected them with curly brackets.

His contribution to the theory of equations is *De aequationum recognitione et emendatione* (1615; *Concerning the Recognition and Emendation of Equations*), in which he presented methods for solving equations of second, third, and fourth degree. Viète knew the connection between the positive roots of an equation (which, in his time, were thought of as the only roots) and the coefficients of the different powers of the unknown quantity.[4] Viéte’s new systematic algebra opened doors for further mathematical development enabling his successors to express their thoughts mathematically with a higher degree of detail and precision.

Jeff Suzuki, *Francois Viete and the Birth of Algebra*, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “François Viète“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Robin Hartshorne: Francois Viete – Life, Mathematicians, at UC Berkeley
- [3] “Viète, François.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com
- [4] François Viète, seigneur de la Bigotiere, French mathematician, at Britannica Online
- [5] Scaliger and the Science of Chronology, SciHi Blog, January 21, 2016.
- [6] Cogito Ergo Sum – René Descartes, SciHi Blog, March 31, 2013.
- [7] François Viète at zbMATH
- [8] François Viète at GND
- [9] François Viète at Wikidata
- [10] François Viète at Reasonator
- [11] Jeff Suzuki,
*Francois Viete and the Birth of Algebra*, Jeff Suzuki: The Random Professor @ youtube - [12] Cantor, Moritz (1911). “Vieta, François“. In Chisholm, Hugh (ed.).
*Encyclopædia Britannica*. Vol. 28 (11th ed.). Cambridge University Press. pp. 57–58. - [13] Francois Viete,
*In artem analyticem isagoge : seorsim excussa ab Opere restitutae mathematicae analyseos, seu Algebra nova*, 1591 - [14] Timeline for François Viète, via Wikidata

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]]>The post Georg Joachim Rheticus and the Copernican Revolution appeared first on SciHi Blog.

]]>On February 16, 1514, mathematician, cartographer, navigational-instrument maker, medical practitioner, and teacher **Georg Joachim Rheticus** was born. He is perhaps best known for his trigonometric tables and as Nicolaus Copernicus’ [4] sole pupil, who facilitated the publication of his master’s famous work *De revolutionibus orbium coelestium* (*On the Revolutions of the Heavenly Spheres*).

Georg Joachim Rheticus was born in Feldkirch, the county of Tyrol in the Holy Roman Empire, as the son of Georg Iserin, the town medical officer of Feldkirch, and the northern Italian noblewoman Thomasina de Porris. He was taught by his father in his early life, but when the young Rheticus was only 14 years old, his father was accused of witchcraft and fraud and executed in 1528. The boy was then highly supported by Achilles Gasser, who took over his father’s medical practice. He helped Rheticus to begin his studies at the Latin school in Feldkirch, continuing his education by studying mathematics in Zurich from 1528 to 1531, and then at the University of Wittenberg, where he earned the degree of Master of Liberal Arts in 1536. [1]

Rheticus was then appointed teacher of mathematics and astronomy at the University of Wittenberg after he received his degree. He was back then supported by Philipp Melanchthon,[6] who also helped Rheticus to study with the leading astronomers at the time. In 1538, Rheticus traveled to Nuremberg where he met famous polymath and instrument maker Johann Schöner [5] as well as the printer Petreius, who presumably commissioned him to persuade Nicolaus Copernicus to publish his magnum opus in Nuremberg. At least Petreius gave him three books from his publishing house as a gift for Copernicus.[16] In Ingolstadt, Rheticus visited mathematician, astronomer and cosmographer Peter Apianus and went on to Tübingen in order to meet humanist and philologist Joachim Camerarius, the rector of the University of Leipzig. In the Spring of 1539, Rheticus arrived at Frombork (Frauenburg) in Ermland where he studied for two years with Copernicus. An experience he remembered as following:

“I heard of the fame of Master Nicolaus Copernicus in the northern lands, and although the University of Wittenberg had made me a Public Professor in those arts, nonetheless, I did not think that I should be content until I had learned something more through the instruction of that man. And I also say that I regret neither the financial expenses nor the long journey nor the remaining hardships. Yet, it seems to me that there came a great reward for these troubles, namely. that I, a rather daring young man compelled this venerable man to share his ideas sooner in this discipline with the whole world.” [2]

In 1539, the *Narratio Prima de libris revolutionum Copernici* *(First report to Johann Schöner on the Books of the Revolutions of the learned gentleman and distinguished mathematician, the Reverend Doctor Nicolaus Copernicus of Toruń, Canon of Warmia, by a certain youth devoted to mathematics)* was published. It is believed that to this day, the work remains the best introduction to Copernicus’s work. Rheticus managed to put himself in favor with the Duke Albert of Prussia and asked successfully for the permission to publish Copernicus’s *De Revolutionibus*. In 1541, Rheticus returned to the University of Wittenberg, where he was elected dean of the Faculty of Arts. In the same year, he published the trigonometrical sections of Copernicus’s *De Revolutionibus*, adding tables of sines and cosines, however, not naming them this way. [1,2] Rheticus was the first to contribute significantly to the spread of the Copernican world system. He was the only student of Copernicus, and during his stay in Frauenburg he was able to convince him to put his main work into print. During this time he published the first communication about the same in the *Narratio Prima de libris revolutionum Copernici*. He had to leave the correction of the proofs of *De revolutionibus* to Andreas Osiander.[9] The latter took out a theological treatise by Rheticus on the compatibility of the heliocentric system with Holy Scripture and anonymously replaced it with a new preface written by him, presenting the new system as a mere computational model.

One year later, Rheticus left Wittenberg for Nuremberg, where he supervised the printing of *De Revolutionibus*, but continued his journey to Leipzig before the work was finished. By 1548 he was on the road again, visiting Gerolamo Cardano [8] in Milan and beginning medical studies in Zurich. Again through Melanchthon’s intercession, he was admitted to the theological faculty in Leipzig. He began his teaching position in Leipzig as the professor of higher mathematics. Rheticus managed to publish a calendar and ephemeris of 1550 and also an ephemeris and calendar of 1551. In 1551 however, the scientist was forced to leave Leipzig because he was suspected of having a homosexual affair with a student. Rheticus was found guilty of raping the son of Hans Meusel, a merchant, though the exact nature of this encounter has been called into question. According to Meusel, Rheticus “*plied him with a strong drink, until he was inebriated; and finally did with violence overcome him and practice upon him the shameful and cruel vice of sodomy*“.[12] He fled following this accusation, for a time residing in Chemnitz before eventually moving on to Prague.Rheticus was then found guilty in his trial in absentia and consequently exiled from Leipzig for 101 years as well as having his possessions impounded. Now even Melanchton turned away from Rheticus.

Rheticus then managed to study medicine in Prague and moved to Kraków where he became a practicing doctor, but he continued with his famous trigonometric tables and made instruments which he used for observations and experiments. He also earned significant merits through his 10-digit tables of trigonometric functions, which progressed from 10 to 10 seconds, but the calculation of which was only completed by his pupil Valentin Otho, who also published them in the *Opus palatinum de triangulis* (Heidelberg 1596). [2,3] Shortly before his death in 1573, Rheticus moved to Kaschau in Upper Hungary, where the imperial governor Hans Rueber zu Pixendorf (1529-1584) took him in.

N. J. Wildberger, Rheticus and 17th century trig tables | WildTrig: Intro to Rational Trigonometry, [16]

**References and Further Reading:**

- [1] Das war Georg Joachim Rheticus
- [2] John J. O’Connor, Edmund F. Robertson:
*Georg Joachim Rheticus.*In:*MacTutor History of Mathematics archive.* - [3] Georg Joachim Rheticus at Britannica
- [4] Nikolaus Copernicus and the Heliocentric Model, SciHi Blog, February 19, 2013.
- [5] Johannes Schöner and his Globes, SciHi Blog, January 16, 2015.
- [6] Philipp Melanchton – the First Systematic Theologician of the Protestant Reformation, SciHi Blog, February 16, 2021.
- [7] Gerolamo Cardano and Physician, Mathematician, and Gambler, SciHi Blog
- [8] Andreas Osiander and Copernicus’ Revolutions, SciHi Blog
- [9] Works by or about Georg Joachim Rheticus at German National Library
- [10] Works by or about Georg Johann Rheticus at Wikisource
- [11] Siegmund Günther (1889), “Rheticus, Georg Joachim“,
*Allgemeine Deutsche Biographie (ADB)*(in German),**28**, Leipzig: Duncker & Humblot, pp. 388–390 - [12] Repcheck, Jack (2008).
*Copernicus’s Secret: How the Scientific Revolution Began*. Simon & Schuster. p. 178. - [13] Georg Joachim Rheticus (1540),
*Narratio Prima*– Full digital facsimile, Linda Hall Library. - [14] Georg Joachim Rheticus at Wikidata
- [15] Timeline of Austrian Astronomers, via Wikidata and DBpedia
- [16] N. J. Wildberger, Rheticus and 17th century trig tables | WildTrig: Intro to Rational Trigonometry, Insights into Mathematics @ youtube

- [17] The emergence of modern astronomy – a complex mosaic: Part IX,
*The Renaissance Mathematicus*, April 24, 2019

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]]>The post Richard Brauer and the Theory of Algebra appeared first on SciHi Blog.

]]>On February 10, 1901, German and American mathematician **Richard Dagobert Brauer** was born. Brauer worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory.

Richard Brauer was born in Charlottenburg, a district of Berlin, Germany, which was not incorporated into the city until 1920, to Max Brauer, a well-off businessman in the wholesale leather trade, and his wife Lilly Caroline. Alfred Brauer was Richard’s older brother and both were interested in science and mathematics, but Alfred was injured in combat in World War I, while Richard was young enough to avoid being drafted into the army. As a boy, Richard dreamt of becoming an inventor, and in February 1919 enrolled in Technische Hochschule Berlin-Charlottenburg. He soon transferred to University of Berlin. Except for the summer of 1920 when he studied at University of Freiburg, he studied in Berlin, being awarded his Ph.D. (with distinction) in 1926. Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result. The problem also was solved by Heinz Hopf at the same time. Richard wrote his thesis under Schur, providing an algebraic approach to irreducible, continuous, finite-dimensional representations of real orthogonal (rotation) groups.

Before the award of his doctorate, however, Brauer had married Ilse Karger in September 1925. They had been a fellow students in one of Schur’s courses on number theory.[1] Their sons George Ulrich (b 1927) and Fred Gunther (b 1932) also became mathematicians. Brauer began his teaching career in Königsberg (now Kaliningrad) working as Konrad Knopp’s assistant. Brauer expounded central division algebras over a perfect field while in Königsberg; the isomorphism classes of such algebras form the elements of the Brauer group he introduced.

Political events forced Brauer’s family to move. When the Nazi Party took over in 1933, the Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer and other Jewish scientists. Brauer was offered an assistant professorship at University of Kentucky. He accepted the offer, and by the end of 1933 he was in Lexington, Kentucky, teaching in English. Ilse followed the next year with George and Fred; brother Alfred made it to the USA in 1939, but their sister Alice was killed in the Holocaust.

Hermann Weyl invited Richard to assist him at Princeton’s Institute for Advanced Study in 1934,[10] where together with Nathan Jacobson, he edited Weyl’s lectures *Structure and Representation of Continuous Groups*. Collaboration between Brauer and Weyl on several projects followed, in particular a famous joint paper on spinors published in 1935 in the *American Journal of Mathematics.* This work was to provide a background for the work of Paul Dirac in his exposition of the theory of the spinning electron within the framework of quantum mechanics.[1,8] Through the influence of Emmy Noether[9], Richard was invited to University of Toronto to take up a faculty position and remained there until 1948, when he left to join the faculty at the University of Michigan. During this time Brauer developed some of his most impressive theories, carrying the work of Georg Frobenius into a whole new setting, in particular the work on group characters Frobenius published in 1896. Brauer carried Frobenius’s theory of ordinary characters, where the characteristic of the field does not divide the order of the group, to the case of modular characters, where the characteristic does divide the group order.[1,4]

Brauer also studied applications to number theory. With his graduate student Cecil J. Nesbitt he developed modular representation theory, published in 1937. Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic. In 1948 Richard and Ilse moved to Ann Arbor, Michigan where he and Robert M. Thrall contributed to the program in modern algebra at University of Michigan. With his graduate student K. A. Fowler, Brauer proved the Brauer-Fowler theorem. In mathematical finite group theory, the Brauer–Fowler theorem, states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g^{1/3}. Donald John Lewis was another of his students at UM.

In 1952 Brauer joined the faculty of Harvard University. He was chairman of the department from 1959 to 1963. The Brauer–Fowler theorem later provided significant impetus towards the classification of finite simple groups, for it implied that there could only be finitely many finite simple groups for which the centralizer of an involution (element of order 2) had a specified structure. Brauer applied modular representation theory to obtain subtle information about group characters, particularly via his three main theorems. These methods were particularly useful in the classification of finite simple groups with low rank Sylow 2-subgroups. The Brauer–Suzuki theorem showed that no finite simple group could have a generalized quaternion Sylow 2-subgroup, and the Alperin–Brauer–Gorenstein theorem classified finite groups with wreathed or quasidihedral Sylow 2-subgroups.

Before retiring in 1971 he taught aspiring mathematicians such as Donald Passman and I. Martin Isaacs. Brauer was to spend the rest of his life working on the problem of classifying the finite simple groups. He died before the classification was complete but his work provided the framework of the classification which was completed only a few years later.[1] Richard Brauer died on April 17, 1977, aged 76.

SummerSchool 20060717 1430 Kresch – Brauer groups, Galois cohomology, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Richard Brauer“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Richard Dagobert Brauer, American mathematician, in Britannica Online
- [3] J. A. Green: Richard Dagobert Brauer, 1901-1977, A Biographical Memoir, National Academy of Sciences
- [4] Ferdinand Georg Frobenius and Group Theory, SciHi blog, August 3, 2016.
- [5] Richard Bauer at zbMATH
- [6] Richard Brauer at Mathematics Genealogy Project
- [7] Richard Brauer at Wikidata
- [8] Paul Dirac and the Quantum Mechanics, SciHi Blog, August 8, 2013.
- [9] Emmy Noether and the Love for Mathematics, SciHi Blog, March 23, 2013.
- [10] Hermann Weyl – between Pure Mathematics and Theoretical Physics, SciHi Blog
- [11] SummerSchool 20060717 1430 Kresch – Brauer groups, Galois cohomology, Graduate Mathematics @ youtube
- [12] Timeline of Group Theorists, via DBpedia & Wikidata

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]]>The post Giambattista della Porta – Natural Magic and the Academy of Secrets appeared first on SciHi Blog.

]]>On February 4, 1615, Italian scholar, polymath and playwright **Giambattista della Porta** passed away. Besides occult philosophy, astrology, alchemy, mathematics, meteorology, and natural philosophy, della Porta worked on cryptography and also on optics. He claimed to be the inventor of the telescope although he does not appear to have constructed one before Galileo [4].

“Having observed the forces of all things natural and celestial and having examined by painstaking investigation the sympathy among those things, brings into the open powers hidden and stored away in nature; thus, magic links lower things (as if they were magical enticements) to the gifts of higher things…so that astonishing miracles thereby occur.”

– Cornelius Agrippa, on natural magic in De occulta philosophia, 1533

Giambattista della Porta was born at Vico Equense, near Naples, in the Kingdom of Naples. He came from a noble family, the third of four sons of Nardo Antonio della Porta. His mother, of Calabrian origin, was a Spadafora, a sister of the scholar Adriano Guglielmo Spadafora, who became curator of the archives of Naples in 1536 and guided the young Giambattista in his studies. However, Giambattista was self-taught and in addition to having talents for the sciences and mathematics, he and his brothers were also extremely interested in the arts, music in particular. His education consisted of the study of classical letters, philosophy and natural sciences, without neglecting music, dance, horseback riding, and gymnastics. But he did not go to university (in fact, the University of Naples was not officially founded until 1581) and always stayed away from academic circles. However, the Neapolitan family home was frequented by philosophers, mathematicians, poets and musicians, forming a real academy.

In his works Giambattista della Porta left contradictory information about his date of birth or the dates of writing of his works. He loved to keep many things about himself secret, and was interested in subjects that sometimes bordered on what the Inquisition would tolerate. According to his first biographer Pompeo Sarnell, from his youth Giambattista composed speeches in Latin and in the vernacular, and excelled in “natural philosophy”. His insatiable curiosity for all mysteries resulted in the publication of works on natural magic, alchemy, astrology, physiognomy, cryptography, agriculture, the art of memory, optics, pneumatics and geometry during his lifetime.

Because of his methodology and relative neutrality, he is considered one of the first natural scientists in the modern sense. From 1558 to 1589 he published in 20 books the *Magia naturalis* (*The natural Magic*), in which he handed down, among other things, one of the few existing witch ointment recipes. An encyclopedia of popular knowledge, on occult and unusual things, the work brought him his first fame. He claimed (falsely) that he was only 15 years old when he wrote it. He mentions for the first time a water drum blower for the production of a continuous air jet for a forge in Nettuno south of Rome. In volume 20 he described the construction of a tethered kite. The work had more than twenty editions in Latin and was later translated into French, Italian, English, German and Dutch.

His training in the refined milieu of the Neapolitan aristocracy, without the constraints of academic rigor, favored the dilettante “experimental” approach he adopted to reveal all the hidden wonders of the world. Della Porta claimed to be able to submit to “experiment” and rational analysis all the strange coincidences he observed, which he blithely generalized into correlations or even causal relationships. Rather than relying on supernatural beings or powers, he sought to study all surprising and marvelous phenomena without preconceptions and to find natural causes for them. But his “experimental” method was so crude that he was not immune to the greatest credulity. For he did not clearly distinguish history from experience. Like many of his contemporaries, what he had read in the ancient world, like Pliny,[6] was as good as what he had seen with his own eyes. In 1589, on the eve of the early modern Scientific Revolution, della Porta became the first person to attack in print, on experimental grounds, the ancient assertion that garlic could disempower magnets. This was an early example of the authority of early authors being replaced by experiment as the backing for a scientific assertion

A society he founded in Naples in 1560 for the study of nature, the *Academia dei Secreti (Academy of Secrets)*, had to be dissolved by order of the Pope. Founded sometime before 1580, the *Otiosi* were one of the first scientific societies in Europe and their aim was to study the “secrets of nature.” Any person applying for membership had to demonstrate they had made a new discovery in the natural sciences.

His fundamental writing on cryptology, entitled *De furtivis literarum notis* (*On the hidden meaning of letters*), which he published in 1563, made him finally famous. In it he described the first known digraphic substitution cipher. Della Porta also invented a method which allowed him to write secret messages on the inside of eggs. During the Spanish Inquisition, some of his friends were imprisoned. At the gate of the prison, everything was checked except for eggs. Della Porta wrote messages on the egg shell using a mixture made of plant pigments and alum. The ink penetrated the egg shell which is semi-porous. When the egg shell was dry, he boiled the egg in hot water and the ink on the outside of the egg was washed away. When the recipient in prison peeled off the shell, the message was revealed once again on the egg white. In 1566, he published a small work on the art of memorizing information, *Arte del ricordare*, in which he gave mnemonic procedures to improve memory.

Around 1570, he further developed the camera obscura, an early precursor of the camera. In a later edition of his Natural Magic, della Porta described this device as having a convex lens. Though he was not the inventor, the popularity of this work helped spread knowledge of it. He compared the shape of the human eye to the lens in his camera obscura, and provided an easily understandable example of how light could bring images into the eye. Della Porta also claimed to have invented the first telescope, but died while preparing the treatise (*De telescopiis*) in support of his claim. His efforts were also overshadowed by Galileo Galilei’s improvement of the telescope in 1609, following its introduction by Lippershey in the Netherlands in 1608.[6] In 1610, he joined the Accademia dei Lincei, the Lincean Academy, one of the oldest and most prestigious European scientific institutions. Galileo Galilei also joined the Academy in 1611 and soon became its intellectual centre.

In later life, della Porta collected rare specimens and grew exotic plants.* Pomarium* (on fruit growing) and *Olivetum* (on olive growing), published respectively in 1583 and 1584, were incorporated into his more comprehensive work on agriculture, *Villae* of 1592. Before that, however, in 1586, his second major work appeared, a medical work entitled *De humana physiognomia*, followed in 1588 by a work on the physiognomy of plants, “*Phytognomica*.” Here he attempted to infer from the appearance and life history of plants their effects. Thus he stated that herbs with yellow sap could cure jaundice, those with rough surfaces skin diseases. This influenced the Swiss eighteenth-century pastor Johann Kaspar Lavater [7] as well as the 19th century criminologist Cesare Lombroso [8]. Della Porta’s private museum was visited by travelers and was one of the first examples of a natural history museum, inspiring the Jesuit Athanasius Kircher to assemble a similar collection in Rome.[9]

Because of his preoccupation with magic and physiognomy he had to answer to the Inquisition in 1592, but the proceedings remained inconsequential for him. In the following years, however, he withdrew from the sciences as a precaution and turned to dramatic literature. His comedy *La Trappolaria*, one of many successful dramatic works, appeared in Bergamo in 1596. During his lifetime he wrote 3 tragedies, 29 comedies and a tragi-comedy, of which only 2 tragedies, 14 comedies and the tragi-comedy have survived.

Giambattista della Porta passed away on February, 4, 1615, at age 79 in Naples.

Sven Dupré –* Secrets and Experiments: Della Porta’s Optics between Reading and Doing (2015.04.08)*, []

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Giambattista della Porta”,
*MacTutor History of Mathematics archive*, University of St Andrews - [2] Giambattista della Porta: De humana physiognomonia libri IIII (Vico Equense, Italy, 1586)
- [3] Works by or about Giambattista della Porta at Internet Archive
- [4] Galileo Galilei and his Telescope, SciHi Blog
- [5] Pliny the Elder and the Destruction of Pompeii, SciHi Blog
- [6] Hans Lippershey and the Telescope, SciHi Blog
- [7] Johann Lavater – Physiognomic Fragments for the Promotion of Human Knowledge and Human Love, SciHi blog
- [8] Cesare Lombroso – The Father of Criminology, SciHi Blog
- [9] Athanasius Kircher – A Man in Search of Universal Knowledge, SciHi Blog
- [10] Natural magick, by John Baptista Porta, a Neapolitane: in twenty books … wherein are set forth all the riches and delights of the natural sciences From the Collections at the Library of Congress
- [11] Giambattista della Porta at Wikidata
- [12] Timeline of Pre 19th century Cryptographers, via DBpedia and Wikidata

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]]>On February 2, 1522, Italian mathematician **Lodovico Ferrari** was born, who was the first to find an algebraic solution to the biquadratic, or quartic equation.

Born in Bologna, Italy, Lodovico’s grandfather, Bartholomew Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, Italy and by chance, he was able to start his career as servant of one of the most famous mathematicians of the time, Gerolamo Cardano [2]. Initially brought up in his father’s house, Lodovico went to live with his uncle Vincent after his father was killed. This Vincent Ferrari had an own son named Luke, a difficult young man, who decided to run away from home and seek employment in Milan. There discovered that famous mathematician Gerolamo Cardano was looking for a servant. Work did not suit Luke much and after working for Cardano for a while he decided that things were better back home and, without telling Cardano, just left his house. Cardano contacted Vincent Ferrari requesting that he send his son back to continue his employment as a servant in his house. Vincent, however, saw his chance to keep his own son at home and offload the responsibility of supporting his cousin Lodovico, so instead of sending Luke back to Cardano in Milan, he sent Lodovico.[1]

At age 14, Lodovico arrived at Cardano’s house in Milan to take over his cousin Luke’s position. Cardano, upon the discovery that the boy could read and write appointed Ludovico as his secretary, who turned out to be an exceptionally gifted young man and he decided to teach him mathematics. By attending Cardano’s lectures, Ferrari learned Latin, Greek, and mathematics. Ferrari aided Cardano on his solutions for quadratic equations and cubic equations, and was mainly responsible for the solution of quartic equations that Cardano published. While still in his teens, Ferrari was able to obtain a prestigious teaching post at the Piatti Foundation in Milan after Cardano resigned from it and recommended him in 1540.[1]

Ferrari discovered the solution of the quartic equation in 1540 with a quite beautiful argument but it relied on the solution of cubic equations so could not be published before the solution of the cubic had been published. However, there was no way to make this public without the breaking the sacred oath made by Cardano to Niccolò Fontana Tartaglia, a famous fellow mathematician. While Cardano was preparing the publication of his *Practica arithmetice* (1539), news reached Cardano that a method of solving the cubic equation of the form x^{3} + ax = b, where a and b are positive, was known to Tartaglia of Brescia. Until then Cardano had accepted Luca Pacioli’s statement in the *Summa de arithmetica, geometria, proportioni et proportionalita* (1494) that the cubic equation could not be solved algebraically.[5] Cardano had contacted Tartaglia, through an intermediary, requesting that Tartaglia’s method to solve the cubic equation could be included in his book. Tartaglia declined this opportunity, stating his intention to publish his formula in a book of his own that he was going to write at a later date. Cardano, accepting this, then asked to be shown the method, promising to keep it secret. Tartaglia, however, refused.[4]

Despairing of ever publishing their ground breaking work, Cardano and Ferrari travelled to Bologna to call upon their mathematical colleague, Annibale della Nave, who had been appointed there on the death of Scipione del Ferro. Cardano and Ferrari satisfied della Nave that they could solve the ubiquitous cosa and cube problem, and della Nave showed them in return the papers of the late del Ferro, proving that Tartaglia indeed was not the first to discover the solution of the cubic.[2]

Thus, Cardano and Ferrari decided to include the solution of the quartic equation in Cardano’s *Ars magna* (1545; “*Great Art*”). The publication of *Ars magna* brought Ferrari into a celebrated controversy with Tartaglia over the solution of the cubic equation. After six printed challenges and counterchallenges, Ferrari and Tartaglia met in Milan on Aug. 10, 1548, for a public mathematical contest, of which Ferrari was declared the winner. This success brought him immediate fame, and he was deluged with offers for various positions. He accepted that from Cardinal Ercole Gonzaga, regent of Mantua, to become supervisor of tax assessments, an appointment that soon made him wealthy.[3]

Lodovico Ferrari eventually retired young and quite rich. He then moved back to his home town of Bologna where he lived with his widowed sister Maddalena to take up a professorship of mathematics at the University of Bologna in 1565. Shortly thereafter, he died on 21 September 1576 of white arsenic poisoning, according to a legend – because of his sister.

Intermediate Algebra Lecture 11.1: Solving Quadratic Equations By Completing the Square, [9]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Lodovico Ferrari”, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Gerolamo Cardano and the Mathematics of Chances, SciHi blog, September 24, 2013.
- [3] Lodovico Ferrari, Italian Mathematician, at Britannica Online
- [4] O’Connor, John J.; Robertson, Edmund F., “Niccolò Fontana Tartaglia“, MacTutor History of Mathematics archive, University of St Andrews.
- [5] “Ferrari, Ludovico.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com.
- [6] News from the world of maths: The eventful life of Lodovico Ferrari, at plus magazine…living mathematics, January 29, 2008.
- [7] Lodovico Ferrari at Wikidata
- [8] Lodovico Ferrari at zbMATH
- [9] Intermediate Algebra Lecture 11.1: Solving Quadratic Equations By Completing the Square, Professor Leonard @ youtube
- [10] Jayawardene, S. A. (1970–1980). “Ferrari, Lodovico”.
*Dictionary of Scientific Biography*. Vol. 4. New York: Charles Scribner’s Sons. pp. 586–8. - [11]
- Gabriella Belloni Speciale:
*Ferrari, Ludovico.*In: Fiorella Bartoccini (Hrsg.):*Dizionario Biografico degli Italiani*(DBI). Band 46:*Feducci–Ferrerio.*Istituto della Enciclopedia Italiana, Rom 1996. - [12] Timeline of Renaissance Mathematicians, via Wikidata

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