On March 23, 1749, French mathematician and astronomer **Pierre Simon marquis de Laplace** was born, whose work was pivotal to the development of mathematical astronomy and statistics. One of his major achievements was the conclusion of the five-volume *Mécanique Céleste *(*Celestial Mechanics*) which translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.

“One sees, from this Essay, that the theory of probabilities is basically just common sense reduced to calculus; it makes one appreciate with exactness that which accurate minds feel with a sort of instinct, often without being able to account for it.”

– Pierre Simon de Laplace, Introduction to Théorie Analytique des Probabilités (1814)

Pierre Simon Laplace, the son of a cider merchant was born in the Normandie, and grew up as a well educated child attending the local school at a Benedictine priory. In later years, he was sent to Caen in order to study theology and philosophy were he found out about his love to mathematics. Laplace quit his studies in 1768 applying to study mathematics under the most famous French mathematician of his time, Jean-Baptiste le Rond d’Alembert.[4] D’Alembert was immediately impressed by the young Laplace and ready to teach and support him.

Already three years later, Laplace taught geometry, trigonometry, analysis and statistics and published several works on game theory, probability theory and other difficult issues to improve his reputation. He was admitted to the Académie Française at only 24 years. Laplace was about to become one of France’s most influential scientists and had the honor to examine the future members of the royal artillery. One of his aspirants was the 16 year old Napoleon Bonaparte, who later offered Laplace a position as his minister of the interior.

During the French Revolution, Laplace was able to continue his research as far as possible and in 1792 he became a member of the Committee for Weights and Measures, which was later responsible for the introduction of the units metres and kilograms. But Laplace had to give up this office, because with the rule of Robespierre, participation in the revolution and hatred of the monarchy became a condition for the work. After Robespierre himself died by the guillotine on 28 July 1794, Laplace returned to Paris and became one of the two examiners for the École polytechnique in December of that year. In 1795 Laplace resumed his work on the Committee for Weights and Measures and became its chairman. In the same year, the academy was re-established with the umbrella organisation Institut de France. Laplace was a founding member and later also president of the institute. He also took over the management of the Paris Observatory and the research department.

After Laplace had voted in 1804 in the Senate for Napoleon’s appointment as emperor, he ennobled him to count in 1806. In the same year Laplace moved to Arcueil, a suburb of Paris, in the house next door to the chemist Claude-Louis Berthollet, with whom he founded the Société d’Arcueil. There, they conducted experiments with other, mostly young scientists. Among these scientists were Jean-Baptiste Biot and Alexander von Humboldt.[5] Through this work, however, he made enemies because he set up a clear research programme, which mainly included his own research priorities, and also carried this out mercilessly. Laplace lost further reputation because he continued to hold on to the particle nature of light, while wave theory was increasingly recognised by Augustine Jean Fresnel.[6]

Laplace created further opponents when he voted for Napoleon’s deposition in 1814 and immediately made himself available for the Bourbon restoration. King Louis XVIII, on the other hand, made Laplace a Pair of France in 1815 and a Marquis in 1817. In 1816 Laplace stopped working at the École polytechnique and became a member of the 40 Immortals of the Académie française.

However, Laplace’s scientific contributions are numerous. In the field of astronomy, he published a work titled *Traité de Mécanique Céleste*, a collected work of all scientific approaches after Newton. In the book series he also demonstrated the mathematical prove for the stability of planetary orbits – due to irregularities in the orbit curves, it was believed at the time that the solar system could collapse – and hypothesized about the possibility of black holes. This work was a great success and used and studied by every astronomer or those willing to be one. Laplace’s second major field of research was probability theory. For Laplace, it was a way out to achieve certain results despite a lack of knowledge. In his two-volume work *Théorie Analytique des Probabilités* (1812), Laplace gave a definition of probability and dealt with dependent and independent events, especially in connection with gambling. He also dealt in the book with the expected value, mortality and life expectancy. The work refuted the thesis that a strict mathematical treatment of probability was not possible. Laplace has always been more a physicist than a mathematician. Mathematics served him only as a means to an end. Today, however, the mathematical methods that Laplace developed and used are much more important than the actual work itself, like the Laplace operator, or the Laplace transform.

Laplace died in Paris in 1827. His brain was removed by his physician, François Magendie, and kept for many years, eventually being displayed in a roving anatomical museum in Britain.

Gilbert Strang, *Laplace Transform: First Order Equation*, [11]

**References and Further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson: Pierre Simon de Laplace. In: MacTutor History of Mathematics archive
- [2] Laplace at Trinity College Dublin
- [3] Laplace Transform
- [4] Jean Baptiste le Rond d’Alembert and the Great Encyclopedy, SciHi Blog, November 16, 2012.
- [5] On the Road with Alexander von Humboldt, SciHi Blog, August 3, 2012.
- [6] Augustin-Jean Fresnel and the Wave Theory of Light, SciHi Blog, March 10, 2018.
- [7] Cardinal Richelieu and the Académie Francaise, SciHi Blog
- [8] Pierre Simon de Laplace at Wikidata
- [9] Pierre Simon de Laplace at zbMATH
- [10] Works of Pierre Simon de Laplace (digitized at Wiki Commons)
- [11] Gilbert Strang,
*Laplace Transform: First Order Equation*, MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015, MIT Open CourseWare @ youtube - [12] Pierre-Simon Laplace at the Mathematics Genealogy Project
- [13] Andoyer, H. (1922). “L’œuvre scientifique de Laplace”.
*Paris*(in French). Paris Payot. - [14] Timeline for Pierre Simon de Laplace via Wikidata

On March 27, 1857, English mathematician and biostatistician **Karl Pearson** was born. Pearson has been credited with establishing the discipline of mathematical statistics. He founded the world’s first university statistics department at University College London in 1911, and contributed significantly to the field of biometrics, meteorology, theories of social Darwinism and eugenics.

“It was Karl Pearson, a man with an unquenchable ambition for scholarly recognition and the kind of drive and determination that had taken Hannibal over the Alps and Marco Polo to China, who recognized the power in Edgeworth’s formulations of Galton’s ideas. Pearson lacked Galton’s originality and Edgeworth’s depth of understanding, but it was his zeal, with a vital assist from G. Udny Yule, that created the methodology and sold it to the world.”

– Stephen M. Stigler (1986) on Karl Pearson, in [11]

Karl Pearson was born as second child into a family from Yorkshire Quakers in Islington, London, UK, to William Pearson, a barrister of the Inner Temple, and his wife Fanny (née Smith). Pearson was educated privately at University College School, after which he went to King’s College, Cambridge in 1876 to study mathematics, where he was taught by Stokes, Maxwell, Cayley, Burnside, and most importantly the outstanding student coach Edward Routh, to graduate in 1879 as Third Wrangler in the Mathematical Tripos.[1]

He then travelled to Germany to study physics at the University of Heidelberg under G. H. Quincke and metaphysics under Kuno Fischer. He next visited the University of Berlin, where he attended the lectures of the famous physiologist Emil du Bois-Reymond on Darwinism. Pearson also studied Roman Law, taught by Bruns and Mommsen, medieval and 16th century German Literature, and Socialism.

Despite being called to the Bar in 1882, Pearson never practiced law. During 1882-84 he lectured around London on a wide variety of topics such as German social life, the influence of Martin Luther, and historical topics. He even proposed himself to Karl Marx as the English translator of the existing volume of *Das Kapital*.[2,4] He also wrote essays, articles and reviews as well as substituting for professors of mathematics at King’s College and University College London.[1]

In 1884 Pearson was appointed professor of applied mathematics and mechanics at University College, London. He taught graphical methods, mainly to engineering students, and this work formed the basis for his original interest in statistics.[2] 1891 saw him also appointed to the professorship of Geometry at Gresham College. There he met Walter Frank Raphael Weldon, a zoologist who had some interesting problems requiring quantitative solutions. Weldon introduced Pearson to Charles Darwin’s cousin Francis Galton,[5] who was interested in aspects of evolution such as heredity and eugenics. Pearson became Galton’s protégé. Pearson was a co-founder, with Weldon and Galton, of the statistical journal *Biometrika*.[1]

In 1892 Pearson published *The Grammar of Science*, in which he argued that the scientific method is essentially descriptive rather than explanatory. Soon he was making the same argument about statistics, emphasizing especially the importance of quantification for biology, medicine, and social science. It was the problem of measuring the effects of natural selection that captivated Pearson and turned statistics into his personal scientific mission.[2] When young Albert Einstein [6] started the Olympia Academy study group in 1902, with his two younger friends, Maurice Solovine and Conrad Habicht, his first reading suggestion was Pearson’s *The Grammar of Science* which covered several themes that were later to become part of the theories of Einstein and other scientists. Pearson asserted that the laws of nature are relative to the perceptive ability of the observer. Irreversibility of natural processes, he claimed, is a purely relative conception. An observer who travels at the exact velocity of light would see an eternal now, or an absence of motion. He speculated that an observer who travelled faster than light would see time reversal, similar to a cinema film being run backwards. Pearson also discussed antimatter, the fourth dimension, and wrinkles in time.

Through his mathematical work and his institution building, Pearson played a leading role in the creation of modern statistics. The basis for his statistical mathematics came from a long tradition of work on the method of least squares approximation, worked out early in the 19th century in order to estimate quantities from repeated astronomical and geodetic measures using probability theory. Pearson drew from these studies in creating a new field whose task it was to manage and make inferences from data in almost every field. His positivistic philosophy of science provided a persuasive justification for statistical reasoning.[2]

Pearson’s work was all-embracing in the wide application and development of mathematical statistics, and encompassed the fields of biology, epidemiology, anthropometry, medicine, psychology and social history. Among those who assisted Pearson in his research were a number of female mathematicians who included Beatrice Mabel Cave-Browne-Cave and Frances Cave-Browne-Cave. He also founded the journal *Annals of Eugenics* (now *Annals of Human Genetics*) in 1925. Pearson’s commitment to socialism and its ideals led him to refuse the offer of being created an OBE (Officer of the Order of the British Empire) in 1920 and also to refuse a knighthood in 1935.

Karl Pearson died on April 27, 1936, aged 79.

Philippe Rigollet, *1. Introduction to Statistics*, [10]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Karl Pearson“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Karl Pearson, British Mathematician, at Britannica Online
- [3] “Pearson, Karl.” Complete Dictionary of Scientific Biography. Encyclopedia.com
- [4] Karl Marx and Das Kapital, SciHi Blog
- [5] Sir Francis Galton – Polymath, SciHi Blog
- [6] How Albert Einstein Revolutionized Physics, SciHi Blog
- [7] Karl Pearson at zbMATH
- [8] Karl Pearson at Mathematics Genealogy Project
- [9] Karl Pearson at Wikidata
- [10] Philippe Rigollet,
*1. Introduction to Statistics*, MIT 18.650 Statistics for Applications, Fall 2016, MIT Open CourseWare @ youtube - [11] Stephen M. Stigler (1986).
*The History of Statistics: The Measurement of Uncertainty Before 1900.*Harvard University Press. p. 266 - [12] Pearson, Karl (1892).
*The Grammar of Science*. London: Walter Scott. Dover Publications, 2004 - [13] Pearson, Karl (1897).
*The Chances of Death and Other Studies in Evolution*, 2 Vol. London: Edward Arnold. - [14] Pearson, Karl (1905).
*On the General Theory of Skew Correlation and Non-linear Regression*. London: Dulau & Co. - [15] Pearson, Karl (1907).
*A First Study of the Statistics of Pulmonary Tuberculosis*. London: Dulau & Co. - [16] Pearson, Karl, & Barrington, Amy (1909).
*A First Study of the Inheritance of Vision and of the Relative Influence of Heredity and Environment on Sight*. London: Dulau & Co. - [17] Timeline for Karl Pearson, via wikidata

On April 23, 1882, German mathematician and physicist **Emmy Noether** was born, who is best known for her groundbreaking contributions to abstract algebra and theoretical physics. Albert Einstein called her the most important woman in the history of mathematics, as she revolutionized the theories of rings, fields, and algebras.

“My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.”

Letter to Helmut Hasse (1931) as quoted in Auguste Dick, Emmy Noether, 1882-1935 (1981) Tr. H. I. Blocher, p. 61.

Emmy Noether came from a wealthy Jewish family. Her father Max Noether held a chair in mathematics at the University of Erlangen. Emmy did not show any particular precociousness in mathematics, but had an interest in music and dancing in her youth. She attended the *Städtische Höhere Töchterschule*. In April 1900 she passed the state examination to become a teacher of English and French at girls’ schools in Ansbach. In 1903 she passed the external Abitur examination at the Royal Realgymnasium – today’s Willstätter-Gymnasium – in Nuremberg. Emmy Noether decided to enroll at the University of Erlangen, but as one of two women at the institution, she was only allowed to audit her classes instead of really participating in them. Noether also finished her graduation at a grammar school in Nuremberg three years later. While restrictions were hard on studying women in Erlangen, Noether attended lectures of famous scientists like Karl Schwarzschild [5] or David Hilbert [6] in Göttingen. After returning to Erlangen, she was allowed to finally study mathematics and taught at the universities’ mathematical institute without payment after her graduation.

David Hilbert had to put great effort into getting Emmy Noether into Göttingen University as privatdozent. Eventually she was allowed to teach at the university despite her sex, but still without any payment. Since the habilitation of women at Prussian universities was prohibited by a decree, the Department of Mathematics and Natural Sciences of the Faculty of Philosophy of the University of Göttingen submitted an official application to the Prussian minister on 26 November 1915, to make an exception for Emmy Noether. However, the application was rejected and Emmy Noether then had no choice but to announce her lectures under the name of Hilbert, whose assistant she acted as.

In these years, she proved the Noether theorem one of the most important contributions to the field of mathematics since the Pythagorean theorem, as many of her male colleagues noted. It proves a relationship between symmetries in physics and conservation principles. This basic result in the theory of relativity was praised by Einstein in a letter to Hilbert when he referred to Noether’s penetrating mathematical thinking. Noether enjoyed a great reputation and delivered her habilitation lecture in 1919 but was not given any salary for her work until 1924 when Noether was appointed a special teaching position in algebra.

A great part in the development of abstract algebra was achieved in the 20th century and Emmy Noether depicted a major influence on the topic with several papers and lectures. Beginning with the year 1920, Noether began publishing works on the ideal theory, defining left and right ideals as a ring followed by another publication, analyzing ascending chain conditions. Noether published *Abstrakter Aufbau der Idealtheorie in algebraischen Zahlkorpern* in 1924. In this paper she gave five conditions on a ring which allowed her to deduce that in such commutative rings every ideal is the unique product of prime ideals.[3]. After these works, Noether had many supporters in the scientific community and several mathematical terms were named after her. During the lectures she gave at university, Noether gave up regular lesson plans and rather used the time for intense discussions, bringing her research forward. Some students paid lots of respects to her methods, others were rather frustrated.

After a long friendship with the mathematician Pavel Alexandrov, Noether decided to continue her work at the Moscow State University in 1928 for some time. There she critically contributed to the development of Galois theory. Emmy Noether’s achievements are numerous. And even though she received several awards for her works she was still not promoted to being a full professor at the university, which caused much frustration along her colleagues who dearly respected her achievements and her personality.

Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zürich in September 1932. Her address to the 1932 Congress was entitled *Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie.* Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras. In 1932 she also received, jointly with Emil Artin,[7] the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge.

In 1933, Emmy Noether received the same letter as many of her Jewish colleagues. She was expelled from her position at the University of Göttingen due to the new Law for the Restoration of the Professional Civil Service. However, she continued her lectures on class field theory secretly in her apartment until starting her job at the University of Oxford and later the Institute for Advanced Study in Princeton.

Emmy Noether passed away on April 14, 1935. At her memorial, many notable mathematicians and friend’s of Noether like Pavel Alexandrov, Bryn Mawr or Hermann Weyl paid their respects. Although she received little recognition in her lifetime considering the remarkable advances that she made, she has been honoured in many ways following her death.[3]

Emmy Noether is one of the founders of modern algebra. Her mathematical profiling developed in cooperation and discussion with Professor Paul Gordan from Erlangen, who also became her doctoral supervisor. He was often called the “King of the Invariants”. The theory of invariants occupied Emmy Noether until 1919 decidedly. In Göttingen, by then a world centre of mathematical research, she built up her own mathematical school. Noether is also ascribed a decisive role in the implementation of abstract algebraic methods in topology.

Georgia Benkart, *Celebrating Emmy Noether*, [12]

**References and Further Reading:**

- [1] Emmy Noether at the San Diego Supercomputer Center Website
- [2] Emmy Noether at the Agnes Scott College Website
- [3] O’Connor, John J.; Robertson, Edmund F., “Emmy Noether“, MacTutor History of Mathematics archive, University of St Andrews.
- [4] Noether’s Theorem at the University of California’s Website
- [5] Karl Schwarzschild and the Event Horizon, SciHi Blog, October 9, 2014.
- [6] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [7] Emil Artin and Algebraic Number Theory, SciHi Blog, March 3, 2017.
- [8] Emmy Noether at zbMATH
- [9] Emmy Noether at Mathematics Genealogy Project
- [10] Emmy Noether at Women in Mathematics
- [11] Emmy Noether at Wikidata
- [12] Georgia Benkart,
*Celebrating Emmy Noether*, Institute for Advanced Study @ youtube

- [13] Emily Conover (12 June 2018). “Emmy Noether changed the face of physics; Noether linked two important concepts in physics: conservation laws and symmetries”.
*Sciencenews.org*. - [14] Rowe, David E.; Koreuber, Mechthild (2020).
*Proving it her way : Emmy Noether, a life in mathematics*. Cham, Switzerland: Springer. - [15] Angier, Natalie (26 March 2012), “The Mighty Mathematician You’ve Never Heard Of”,
*The New York Times* - [16] Phillips, Lee (May 2015). “The female mathematician who changed the course of physics—but couldn’t get a job”.
*Ars Technica*. California: Condé Nast. - [17] Timeline for Emmy Noether, via Wikidata

On March 11, 1780, German mathematician and civil engineer **August Leopold Crelle** was born. Crelle is best known for being the founder of *Journal für die reine und angewandte Mathematik* (also known as Crelle’s Journal). He also worked on the construction and planning of roads and the first railway in Germany, which was completed in 1838.

“The real purpose of mathematics is to be the means to illuminate reason and to exercise spiritual forces.”

– August Leopold Crelle

August Crelle was born in Eichwerder near Wriezen, Prussia, today on the eastern border of Germany to Poland. His father Christian Gottfried Crelle was an impoverished dike reeve and master builder, who had little in the way of income to be able to give his son a good education. Crelle was therefore largely self-taught, studying civil engineering. Had his family had the resources, then Crelle would have studied mathematics at university. He always had a love of mathematics but earning money was a necessity for him. He succeeded to secure a job as a civil engineer in the service of the Prussian Government and worked for the Prussian Ministry of the Interior on the construction and planning of roads and the one of the first railways in Germany (completed in 1838) between Berlin and Potsdam.[1] He finally obtained the rank of *Geheimer Oberbaurat* and was made a member of the Oberbaudirektion, under the Prussian Ministry of the Interior.[3]

He spent a great deal of time working on mathematics and achieved a remarkable level considering that he had never been formally taught, and when he was 36 years old he submitted a thesis *De calculi variabilium in geometria et arte mechanica usu* to the University of Heidelberg and was duly awarded a doctorate.[1] That same year in a little book *Über einiger Eigenschaften des ebenen geradlinigen Dreiecks* (*On Some Properties of Plane Rectilinear Triangles*), Crelle demonstrated how to determine a point in the interior of a triangle such that the lines joining this point to the vertices of the triangle and the sides of the triangle make equal angles.[4]

Crelle founded a journal devoted entirely to mathematics *Journal für die reine und angewandte Mathematik* (*Journal for Pure and Applied Mathematics*) in 1826, which is often referred to as *Crelle’s Journal* and which he edited until his death. Although not the first such journal, that honor goes to the *Annales de Mathématiques Pures et Appliquées*, founded by Joseph-Diaz Gergonne in 1810, which lasted until 1831. However, Crelle’s was the first periodical devoted exclusively to mathematical research.[4] It was organised quite differently from journals that existed at that time since these other journals were basically reporting meetings of Academies and Learned Societies where papers were read. Crelle was very much in control of the journal, and he acted as editor-in-chief for the first 52 volumes.[1] His journal introduced the work of many new young mathematicians and promoted mathematics instruction. Mathematicians such as Dirichlet, Eisenstein, Grassmann, Hesse, Jacobi, Kummer, Lobachevsky, Möbius, Plücker, von Staudt, Steiner, and Weierstrass all had their early works made famous by publication in Crelle’s journal.[4]

Crelle had the strong ability to spot exceptional talent in young mathematicians. He realised the importance of Norwegian mathematician Niels Henrik Abel‘s work and published several articles by him in this first volume, including his proof of the insolubility of the quintic equation by radicals.[5] Without Crelle, Abel would have remained unknown to the world due to his early death with only 26 years. But Abel was only one of many young mathematicians who benefited from acquaintance with the mathematical enthusiast of limited ability. Crelle also mentored the self-taught Swiss mathematical genius Jakob Steiner, often called “*the greatest geometer since Apollonius.*” [4]

In 1828 Crelle transferred from the Ministry of the Interior to the Ministry of Education, where he was employed as advisor on mathematics, particularly on the teaching of mathematics in high schools, technical high schools, and teachers colleges. During the summer of 1830, on an official tour to France, he studied the French methods of teaching mathematics. In his report to the ministry Crelle praised the organization of mathematical education in France but criticized the heavy emphasis on applied mathematics. In line with the neo-humanistic ideals then current in Germany, he maintained that the true purpose of mathematical teaching lies in the enlightenment of the human mind and the development of rational thinking.[3]

However, he became keen to bring the model of the École Polytechnique to Germany for this was the French route to train high quality teachers. One of the outcomes of his involvement with teaching of mathematics in schools was that he published a large number of textbooks and published multiplication tables that went through many editions. Crelle was elected to the Berlin Academy in 1827 with the strong support of Alexander von Humboldt.[9] Crelle wrote many mathematical and technical papers, textbooks, and mathematical tables and translated works by Lagrange and Legendre. Except for his *Rechentafeln*, these are now mostly forgotten. Also, for many years he published *Journal für die Baukunst*. Although beginning in the 1830’s his health declined until he was hardly able to walk, Crelle continued to further the course of mathematics, even at great personal sacrifice.[3]

Keith Devlin, Raphael Bousso, Philip Clayton, Steven Strogatz, W. Hugh Woodin: *Infinity: The Science of Endless*, [12]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “August Leopold Crelle“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Peter Roquette: August Leopold Crelle (1780-1855) [in German], Rede bei der Aufstellung einer Gedenktafel

in Eichwerder, Crelles Geburtsort, am 18.11.2000 - [3] “Crelle, August Leopold.” Complete Dictionary of Scientific Biography. . Encyclopedia.com
- [4] August Leopold Crelle [pdf], in Robert A. Nowlan, A Chronicle of Mathematical People
- [5] The Short but Influential Life of Niels Henrik Abel, SciHi blog, April 6, 2013.
- [6] August Leopold Crelle at zbMATH
- [7] August Leopold Crelle at Mathematics Genealogy Project
- [8] August Leopold Crelle at Wikidata
- [9] On the Road with Alexander von Humboldt, SciH Blog, August 3, 2018.
- [10] Moritz Cantor:
*Crelle, August Leopold.*In:*Allgemeine Deutsche Biographie*(ADB). Band 4, Duncker & Humblot, Leipzig 1876, S. 589 f. - [11] Wolfgang Eccarius: Der Techniker und Mathematiker August Leopold Crelle (1780–1855) und sein Beitrag zur Förderung und Entwicklung der Mathematik im Deutschland des 19. Jahrhunderts. Dissertation. Eisenach 1974.
- [12] Keith Devlin, Raphael Bousso, Philip Clayton, Steven Strogatz, W. Hugh Woodin:
*Infinity: The Science of Endless*, World Science Festival @ youtube

- [13] Timeline for 19th century German mathematicians, via DBpedia and Wikidata

On March 3, 1845, German mathematician **Georg Cantor**, creator of the set theory was born. **Set Theory** is considered the fundamental theory of mathematics. He also proved that the real numbers are “more numerous” than the natural numbers, which was quite shocking for his contemporaries that there should be different numbers of infinity.

“In mathematics the art of asking questions is more valuable than solving problems.”

– Georg Cantor, Doctoral thesis (1867)

Georg Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia. His father, Georg Waldemar Cantor, was a successful merchant, working as a wholesaling agent and later as a broker in the St Petersburg Stock Exchange. When his father became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated from the Realschule in Darmstadt with an outstanding report, which mentioned in particular his exceptional skills in mathematics, in particular trigonometry. In 1862, Cantor entered the University of Zürich to study mathematics. After receiving a substantial inheritance upon his father’s death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker [3], Karl Weierstrass [4] and Ernst Kummer [5]. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. In 1867, Cantor completed his dissertation in number theory with the title “*De aequationibus secundi gradus indeterminatis*“, under the supervision of Eduard Kummer and Karl Weierstrass [7], at the University of Berlin.

Cantor first taught at a girl’s school in Berlin and in 1868, he joined the Schellbach Seminar for mathematics teachers. During this time he worked on his habilitation and, immediately after being appointed to Halle in 1869, where he spent his entire career, he presented his thesis, again on number theory, and received his habilitation. In 1872 Cantor was promoted to Extraordinary Professor at Halle and 1879 became Full Professor.

Cantor’s work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets, which are easy to understand, and “the infinite”, which was considered a topic for philosophical, rather than mathematical, discussion. Cantor succeeded in proving that there are many possible sizes for infinite sets – even infinitely many. Thereby, he established that set theory was anything else but trivial, and that it needed to be studied. In the course of this study, set theory has become a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects from all areas of mathematics in a single theory, and provides a standard set of axioms to prove or disprove them.

“A set is a Many that allows itself to be thought of as a One.”

– Georg Cantor, as quoted in Infinity and the Mind (1995) by Rudy Rucker.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor’s 1874 paper, “*Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” (“On a Property of the Collection of All Real Algebraic Numbers”)*. This paper was the first to provide a rigorous proof that there was more than one kind of infinity. A first step towards Cantor’s set theory already was his 1873 proof that the rational numbers are countable, i.e. they may be placed in one-one correspondence with the natural numbers – and therefore their count is equal. However, when he tried to extend his proof to real numbers, he was facing difficulties. To decide whether the real numbers were countable proved much harder. But in December 1873 he succeeded with the counter argument and proved that the real numbers were not countable. Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous, i.e. having the same number of elements.

In 1891, he published a paper containing his elegant “diagonal argument” for the existence of an uncountable set. He applied the same idea to prove Cantor’s theorem: t*he cardinality of the power set of a set A is strictly larger than the cardinality of A*. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel‘s first incompleteness theorem.[9]

Also related with the problem of infinite sets is the famous continuum hypothesis – *There is no set whose cardinality is strictly between that of the integers and the real numbers* – introduced by Cantor in 1878. It was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris…but this is already another story [8].

“The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.”

– Georg Cantor, as quoted in Infinity and the Mind (1995) by Rudy Rucker.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918.

David Hilbert described Cantor’s work as [1]:

…the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

How big is infinity? – Dennis Wildfogel, [10]

**References and Further Reading:**

- [1] Georg Ferdinand Ludwig Philipp Cantor at The MacTutor History of Mathematics
- [2] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi Blog, December 7, 2014.
- [3] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog, February 19, 2018.
- [4] Ernst Kummer and his Achievements in Mathematics, SciHi Blog, January 19, 2015.
- [5] Georg Cantor at Wikidata
- [6] Georg Cantor at zbMATH
- [7] Georg Cantor at Mathematics Genealogy Project
- [8] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [9] Kurt Gödel Shaking the Very Foundations of Mathematics, SciHi Blog, April 28, 2012.
- [10] How big is infinity? – Dennis Wildfogel, TED-Ed @ youtube
- [11] Dauben, Joseph Warren (1979).
*Georg Cantor His Mathematics and Philosophy of the Infinite*. princeton university press. - [12] Dauben, Joseph (2004) [1993].
*Georg Cantor and the Battle for Transfinite Set Theory*. Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22 - [13] Zermelo, Ernst (1908). “Untersuchungen über die Grundlagen der Mengenlehre I”.
*Mathematische Annalen*.**65**(2): 261–281. - [14] Cantor, Georg (1895). “Beiträge zur Begründung der transfiniten Mengenlehre (1)”.
*Mathematische Annalen*.**46**(4): 481–512. - [15] Cantor, Georg (1897). “Beiträge zur Begründung der transfiniten Mengenlehre (2)”.
*Mathematische Annalen*.**49**(2): 207–246. - [16] Timeline for Georg Cantor, via Wikidata

On February 16, 1698, French mathematician, geophysicist, geodesist, and astronomer**Pierre Bouguer** was born. In 1735 Bouguer sailed with Charles Marie de La Condamine on a scientific mission to Peru, in order to measure a degree of the meridian arc near the equator. He is also known as “*the father of naval architecture*” and the “*father of photometry*“.

Pierre Bouguer was born in Le Croisic at the French Atlantic coast. He was educated in mathematics and hydrography by his father Jean Bouguer, who was a Royal Professor of Hydrography. It is believed that Bouguer was a real child prodigy who had a deep understanding of mathematics at the age of only 15. In the same year Jean passed away and a new professor of hydrography was sought. Pierre applied for the position and due to his brilliance and great knowledge, he was appointed.

Pierre Bouguer had a great career ahead, winning the Prix of the Académie Royale des Sciences for the first time in 1727 for his work on masts of ships. The second was awarded to Bouguer for his work on the altitudes or stars at the sea and he earned the third for his research on the magnetic declination at sea.

In geodesy and geophysics he is best known for the gravity anomalies named after him, for the first investigations of the plumb line deviation as well as the vertical gradient and the large degree measurement of long meridian arcs in South America. The latter took place from 1735 to 1741 in the then viceroyalty of New Granada (today’s Peru) at the instigation of the Paris Academy (Académie des sciences). With the expedition to Swedish Lapland in 1736 by Pierre-Louis Moreau de Maupertuis, Alexis-Claude Clairaut, Anders Celsius and others, its aim was to clarify the question of whether the Earth’s polar radius is larger than the equatorial radius, as shown by some older and contemporary measurements, including those by Giovanni Domenico Cassini, or vice versa, as would be expected according to Newton’s theory; these measurements were also to be significant for the later definition of the metre. Bouguer’s colleagues were Louis Godin (leader of the expedition) and Charles Marie de La Condamine. However, considerable tensions arose between the three scientists, which eventually led to the splitting of the expedition team. Grade measurements along a profile extending from a little north of Quito to a little south of Cuenca showed that the length of a longitude of the earth at the equator was 56753 toisen (110.612 km) and the equatorial radius of the earth was 3281013 toisen (6394.694 km). With the Arctic measurements of Maupertuis, the flattening of the earth was 1:179 (modern value 1:298.25), but was improved to 1:305 a few years later by Cassini’s measurements in France.

It is believed that Bouguer was one of the first to attempt to measure the Earth’s density with the help of the ‘deflection of a plumb line due to the attraction of a mountain’. Pierre Bouguer published his research results in *La Figure de la terre*. La Condamine and Bouguer had a bitter dispute about the results of the expedition for a long time, which only ended with Bouguer’s death.

Through his investigations into the intensity of light Bouguer became the founder of photometry. He developed the theory that even though the eye could not detect the ‘amount’ of brightness very well, it could indeed detect if two objects had the same brightness. Therefore, he proceeded to compare the brightness of the moon to a candlelight. That method along with Kepler‘s inverse square law was then used to measure brightness. He wrote a description of these investigations in the *Essai d’optique, sur la gradation de la lumière* (Paris 1729) and even more extensively in the *Traité d’optique sur la gradation de la lumière* which was not published by Lacaille until after his death in 1760.Through this work, Bouguer’s law became famous, expressing ‘the relationship between the absorption of radiant energy and the absorbing medium’. Bouguer also invented the heliometer in 1748.

In 1746 he published the first treatise of naval architecture, *Traité du navire*, which among other achievements first explained the use of the metacenter as a measure of ships’ stability. His later writings were nearly all upon the theory of navigation and naval architecture. In January 1750 he was elected a Fellow of the Royal Society. In the field of geodesy he wrote the work *Traité de navigation* (Paris, 1753), which was substantially supplemented by Lacaille in the second (1769) and by Lalande in the third edition (1792). He made his first observations at all in the vicinity of the Chimborazo river by the deviation of the lead solder in the earth’s gravity field due to the attraction of the mountains and the height of the snow line. Pierre Bouguer died on 15 August 1758 in Paris at age 60.

His name is also recalled as the meteorological term Bouguer’s halo (also known as Ulloa’s halo, after Antonio de Ulloa, a Spanish member of his South American expedition) which an observer may see infrequently in fog when sun breaks through (for example, on a mountain) and looks down-sun—effectively a “Fog bow” (as opposed to a “rain-bow”). An infrequently observed meteorological phenomenon; a faint white, circular arc or complete ring of light that has a radius of 39 degrees and is centred on the antisolar point. When observed, it is usually in the form of a separate outer ring around an anticorona. The term Bouguer anomaly, referring to small regional variations in the Earth’s gravity field resulting from density variations in underlying rocks, is named after him.

Radiometry and Photometry – LED Fundamental Series by OSRAM Opto Semiconductors, [7]

**References and Further Reading:**

- [1] Pierre Bouguer at MacTutor History of Mathematics archive by O’Connor and Robertson
- [2] Pierre Bouguer,
*Essai d’Optique, sur la gradation de la lumiere*(Paris, France: Claude Jombert, 1729) - [3] Pierre Bouguer at Britannica
- [4] Pierre Bouguer at Wikidata
- [5] Chisholm, Hugh, ed. (1911). “Bouguer, Pierre“.
*Encyclopædia Britannica*(11th ed.). Cambridge University Press. - [6] Ferreiro, Larrie (2011).
*Measure of the Earth: The Enlightenment Expedition that Reshaped Our World*. New York: Basic Books - [7] Radiometry and Photometry – LED Fundamental Series by OSRAM Opto Semiconductors, OSRAM Opto Semiconductors @ youtube
- [8] Lamontagne, Roland. “
*Pierre Bouguer, 1698–1758, un Blaise Pascal du XVIIIe siècle; Suivi d’une correspondence**(Pierre Bouguer, 1698–1758, a Blaise Pascal of the 18th century; followed by correspondence)*“. Manuscript. Montreal: Université de Montreal, 1998 - [9] Pierre Bouguer,
*Traité du navire, de sa construction et de ses mouvemens*. Paris 1746 - [10] Timeline of
**Geodesists****,**via DBpedia and Wikidata

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

“Beware of bugs in the above code; I have only proved it correct, not tried it.”

– Donald Knuth (1977)

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

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

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

“I can’t be as confident about computer science as I can about biology. Biology easily has 500 years of exciting problems to work on. It’s at that level.”

– Donald Knuth (1993)

In the autumn of 1960 Knuth entered the California Institute of Technology and, in June 1963, he was awarded a Ph.D. in mathematics for his thesis ‘*Finite semifields and projective planes*‘. In fact in addition to the work for his doctorate in mathematics, Knuth had from 1960 begun to put his very considerable computing expertise to uses other than writing papers becoming a software development consultant to the Burroughs Corporation in Pasadena, California. Besides, knowledge of his computing expertise was so well established by 1962 that, although he was still a doctoral student at the time, Addison-Wesley approached him and asked him to write a text on compilers.

He began to work at CalTech as associate professor and the commission from Addyson-Wesley turned out into the writing of his seminal multivolume book ‘*The Art of Computer Programming*‘. This work was originally planned to be a single book, and then planned as a six- and then seven-volume series. Originally, the publisher had commissioned Knuth, who was still a master student by that time, to write a single book about compilers. However, Knuth wanted to present all the necessary knowledge on this subject in a mature form. After completing his studies, he wrote to the publisher asking permission to describe things in more detail. The first handwritten draft from 1967 comprised 3900 pages. This led to the plan to write a seven-part series covering the essential basics of computer programming. In 1968, just before he published the first volume, Knuth was appointed as Professor of Computer Science at Stanford University.

As of 2012, the first three volumes and part one of volume four of his series have been published. At the end of each chapter there is a section on history and bibliography with historical information. The exercises are divided into levels of difficulty (and marked accordingly), ranging from extremely simple (00) to the unresolved research problem (50).

“Science is what we understand well enough to explain to a computer. Art is everything else we do.”

– Donald Knuth (1996)

After producing the third volume of his book series in 1976, he expressed such frustration with the nascent state of the then newly-developed electronic publishing tools (especially those that provided input to phototypesetters) that he took time out to work on typesetting and created the *TeX* and *METAFONT* tools. Originally, the artwork for the book was set with the Monotype technique. This type of formula set was complex. After the publication of volume 3 in 1973, Knuth’s publisher sold its Monotype machines. The corrected new editions of Volumes 1 and 3, which appeared in 1975, had to be set in Europe, where some Monotype systems were still in use. The new edition of Volume 2 was to be produced in 1976 with phototypesetting, but the quality of the first samples disappointed Knuth. He had put 15 years of work into the series and only wanted to continue them if the books were set accordingly well. In February 1977 there was a way out when Knuth was presented with the output of a digital printing system with 1000 dpi resolution as part of a book evaluation. When Knuth realized this, he interrupted work on Volume 4, of which he had completed the first 100 pages, and decided to write the programs himself that he and his publisher needed to re-set Volume 2. TeX’s design began on May 5, 1977. METAFONT in addition to TeX, is an abstract description language for the definition of vector fonts. One of the characteristics of Metafont is that all of the shapes of the glyphs are defined with geometrical equations. In particular, one can define a given point to be the intersection of a line segment and a Bézier cubic.

In addition to Knuth’s efforts to achieve an appealing aesthetic appearance in text typesetting, correctness is his primary concern. For this reason, he gives a reward of a “hexadecimal dollar” worth $2.56 (100 hexadecimal corresponds to 256 decimal) for every newly found error in his books or programs. Very few of these checks have been cashed so far.[5] Since Knuth no longer considers checks to be safe, the coveted recognition checks have been issued since 2008 as personal deposits at the fictitious bank of San Serriffe.

“Science is knowledge which we understand so well that we can teach it to a computer; and if we don’t fully understand something, it is an art to deal with it.”

– Donald Knuth, Computer Programming as an Art (1974)

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

In the autumn of 1999 Knuth gave six lectures at MIT on cross-connections between computer science and religion from his personal point of view as part of a series of lectures on “*God and computers*” held by prominent scientists over several years, and took part in a concluding panel discussion. Their notes were published in his book Things a computer scientist rarely talks about. He is a multiple honorary doctor; from 1980 to 2005 he was awarded 25 honorary doctorates, including from ETH Zurich (2005) and Eberhard Karls University of Tübingen (2001).

TeX has changed the technology of mathematics and science publishing since it enables mathematicians and scientists to produce the highest quality of printing of mathematical articles yet this can be achieved simply using a home computer. However, it has not only changed the way that mathematical and scientific articles are published but also in the way that they are communicated.[5] Knuth has been retired since 1993 to dedicate himself exclusively to the completion of *The Art of Computer Programming*. Since February 2011 volume 4A is available, which deals with combinatorics. Volumes 4B and 4C will follow, volume 5 (of seven planned) he hopes to finish by 2025.

Donald Knuth, *The Analysis of Algorithms (2015, recreating 1969)*, [8]

**References and Further Reading:**

- [1] Donald Knuth at Mac Tutor’s History of Mathematics
- [2] Donald Knuth at wikipedia
- [3] Donald E. Knuth,
*The Art of Computer Programming*, Volumes 1–4, Addison-Wesley Professional - [4] Donald Knuth at zbMATH
- [5] O’Connor, John J.; Robertson, Edmund F., “Donald Knuth“, MacTutor History of Mathematics archive, University of St Andrews.
- [6] Donald Knuth at Wikidata
- [7] Scholia entry for Donald Knuth
- [8] Donald Knuth,
*The Analysis of Algorithms (2015, recreating 1969)*, stanfordonline @ youtube - [9] Donald Ervin Knuth – Stanford Lectures (Archive)
- [10] Siobhan Roberts, The Yoda of Silicon Valley.
*The New York Times*, 17 December 2018. - [11] Timeline of works by Donald Knuth, via Wikidata

On January 7, 1871, French mathematician **Félix Édouard Justin Émile Borel** was born. Borel is known for his founding work in the areas of measure theory and probability. In one of his books on probability, he proposed the thought experiment that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque nationale de France (National Library). This is now popularly known as the infinite monkey theorem.

“Whatever the progress of human knowledge, there will always be room for ignorance, hence for chance and probability.”

Emile Borel (1914). Le hasard. Librairie Félix Alcan. p. 12-13.

Émile Borel was born in Saint-Affrique, Aveyron, France, the son of a Protestant pastor, and exhibited his mathematical talent from a young age. He studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure and the École Polytechnique. Although placing first in the 1889 entrance exams for both, he chose to attend École normale supérieure. While studying at university he undertook military service with the engineers at Montpellier.[1] After graduating in 1892 as first in his class, he placed first in the agrégation, a competitive civil service examination leading to the position of professeur agrégé. His thesis, published in 1893, was titled *On some points in the theory of functions*.

When still only 22 years of age, Borel was appointed Maître de Conférence at the University of Lille, and during his four-year stay he published 22 research papers. He returned to Paris in January 1897 when appointed Maître de Conférence at the École Normale Supérieure. From 1899 to 1902 he taught at the Collège de France and was reserve for the Cours Peccot. He was appointed examiner for entry to the École Navale in 1900, holding this position for ten years. He was awarded the Grand Prix of the Academy of Sciences in 1898, he was awarded the Poncelet Prize in 1901, he received the Vaillant Prize in 1904, and in 1905 he was elected president of the French Mathematical Society.[1] In 1909 Borel was appointed to a chair of Theory of Functions created specially for him at the Sorbonne and he went on to hold this professorship until 1941.

In 1913, Émile Borel published the article “*Mécanique Statistique et Irréversibilité*” (*Statistical mechanics and irreversibility*,[12] as well as in his book “*Le Hasard*” in 1914),[13] in which he introduced the amusing thought experiment that entered popular culture under the name “infinite monkey theorem” or the like. The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.[6] In fact the monkey would almost surely type every possible finite text an infinite number of times. However, the probability of a universe full of monkeys typing a complete work such as Shakespeare’s Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero).

In 1939, Jorge Luis Borges wrote an essay called “*The Total Library*” [14] in which he traced the idea of an infinite number of monkeys randomly typing on a keyboard eventually producing all the books in the British Museum back to Aristotle’s comparison of the arrangement of atoms to the arrangement of letters in a tragedy or comedy. Borges followed the concept through the works of Blaise Pascal and Jonathan Swift and then imagined the contents of the total library that would result from the monkeys’ typing, including everything that has ever been written or could be written and a vast amount of meaningless cacophony. Borges expanded on this idea in his widely read 1941 short story “*The Library of Babel*“, which describes a library containing every possible book that could be composed from the letters of the alphabet.[3]

In 1921 Borel was elected to the Académie des Sciences, becoming its vice-president in 1933 and its president in 1934. In 1928, with financial support from Rockefeller and Rothschild, he set up the Institut Henri Poincaré [4] (the Centre Émile Borel is now part of the Institute) and he ran the Institute for thirty years. Along with René-Louis Baire and Henri Lebesgue,[5] Émile Borel was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his honor. He also published a series of papers (1921–27) that first defined games of strategy. In 1913 he was able to bridge the gap between hyperbolic geometry and special relativity with expository work.[1] Borel discovered the elementary proof of Picard’s theorem. This sensational accomplishment set the stage for his formulation of a theory of entire functions and the distribution of their values, a topic that dominated the theory of complex functions for the next 30 years.[2]

Borel also served in the War Office during World War I, in the French Chamber of Deputies (1924–36), and as minister of the navy (1925–40). After his arrest and brief imprisonment under the Vichy regime during World War II, he returned to his native village and worked in the Resistance.[2] Émile Borel died in Paris on 3 February 1956, at age 85.

N J Wildberger, *Emile Borel: Real number enthusiast or skeptic? | Sociology and Pure Mathematics*, [11]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Émile Borel“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Emile Borel, French mathematician, at Britannica online
- [3] Jorge Luis Borges and the Library of Babel, SciHi blog, August 24, 2012
- [4] Henri Poincaré – the Last Universalist of Mathematics, SciHi Blog
- [5] Henri Léon Lebesgue and the Theory of Integration, SciHi Blog
- [6] Brush Up Your Shakespeare, SciHi Blog
- [7] Works by or about Émile Borel at Internet Archive
- [8] Emile Borel at zbMATH
- [9] Emile Borel at Wikidata
- [10] Timeline for Emile Borel, via Wikidata
- [11] N J WIldberger,
*Emile Borel: Real number enthusiast or skeptic? | Sociology and Pure Mathematics*, Insights into Mathematics @ youtube - [12] Émile Borel (1913). “Mécanique Statistique et Irréversibilité”.
*J. Phys. (Paris)*. Series 5.**3**: 189–196. - [13] Émile Borel (1914).
*La hasard*. F. Alcan - [14] Borges, Jorge Luis (August 1939). “
*La biblioteca total” [The Total Library]*. Sur. No. 59. republished in Selected Non-Fictions. - [15] von Neumann, J.; Fréchet, M. (1953). “Communication on the Borel Notes”.
*Econometrica*.**21**(1): 124–127. - [16] Emile Borel at Mathematics Genealogy Project

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

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

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

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

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

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

The ‘method of fluxions’, as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions.[13] In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his *Principia Mathematica* (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.

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

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

— Isaac Newton, Philosophiae Naturalis Principia Mathematica, Third edition

Due to a dispute with his fellow scientist Robert Hooke,[17] who claimed that Newton had stolen some of his optical results, Newton turned in on himself and away from the Royal Society which he associated with Hooke as one of its leaders. He even delayed the publication of a full account of his optical researches until after the death of Hooke in 1703. Newton’s *Opticks* appeared in 1704. In 1687, with the support of his friend the astronomer Edmond Halley [6], Newton published his single greatest work, the ‘*Philosophiae Naturalis Principia Mathematica*‘, in which he showed how a universal force, gravity, applied to all objects in all parts of the universe. The *Principia* is recognized as the greatest scientific book ever written. Newton analyzed the motion of bodies in resisting and non-resisting media under the action of centripetal forces. The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalized that all heavenly bodies mutually attract one another.[13]

In 1689, Newton was elected member of parliament for Cambridge University and in 1696, by the support of his friend and ex-student, Charles Montagu, 1st Earl of Halifax, Newton was appointed warden of the Royal Mint, settling in London. He took his duties at the Mint very seriously and campaigned against corruption and inefficiency within the organisation. As a scholar, Newton held court in the fashionable London coffee houses, surrounded by his acolytes, for whom the term Newtonians was originally minted, handing out unpublished manuscripts to the favored few for their perusal and edification [15]. Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed’s *Historia Coelestis Britannica*, which Newton had used in his studies.[16]

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. Newton was referred to being a rather difficult man, prone to depression and often involved in bitter arguments with other scientists, but by the early 1700s he was the dominant figure in British and European science. Isaac Newton died on 31 March 1727 [NS] (20 March 1726 [OS]) at age 84 and was buried in Westminster Abbey.

Robin Wilson and Raymond Flood, *The World of Isaac Newton*, [18]

**References and Further Reading:**

- [1] Isaac Newotn ond his famous Principia, at SciHi blog
- [2] Isaac Newton at biography.com
- [3] Man is Man’s Wolf – Thomas Hobbes and his Leviathan, SciHi Blog, December4, 2012.
- [4] Dr. Robert A. Hatch: Sir Isaac Newton
- [5] Isaac Newton at BBC History
- [6] Edmond Halley besides the Eponymous Comet, SciHi Blog, November 8, 2015.
- [7] Sir Edmond Halley and his Famous Comet, SciHi Blog, November 8. 2012.
- [8] Cogito Ergo Sum – René Descartes, SciHi Blog, March 31, 2013.
- [9] Pierre Gassendi and his Trials to reconcile Epicurean atomism with Christianity, January 22, 2015.
- [10] Leibniz and the Integral Calculus, SciHi Blog, November 11, 2013.
- [11] Sir Isaac Newton at Wikidata
- [12] Isaac Newton at zbMATH
- [13] O’Connor, John J.; Robertson, Edmund F., “Sir Isaac Newton“, MacTutor History of Mathematics archive, University of St Andrews.
- [14] Isaac Newton at Mathematics Genealogy Project
- [15] Christmas Trilogy 2017 Part 1: Isaac the Imperator, at The Renaissance Mathematicus, Dec 25, 2017.
- [16] John Flamsteed – Astronomer Royal, SciHi Blog
- [17] Robert Hooke and his Famous Observations of the Micrographia, SciHi Blog
- [18] Robin Wilson and Raymond Flood,
*The World of Isaac Newton*, Gresham College @ youtube - [19] Timeline for Isaac Newton, via Wikidata

On December 13, 1557, Italian Renaissance mathematician and engineer **Niccolò Fontana Tartaglia** passed away. Tartaglia is best known today for his contributions in solving cubic equations. He published many books, including the first Italian translations of Archimedes [1] and Euclid,[2] and an acclaimed compilation of mathematics.

“When the cube and the things together

Are equal to some discrete number,

Find two other numbers differing in this one.

Then you will keep this as a habit

That their product shall always be equal

Exactly to the cube of a third of the things.

The remainder then as a general rule

Of their cube roots subtracted

Will be equal to your principal thing. “

– Niccolò Tartaglia, the poem in which he revealed the secret of solving the cubic to Cardano

In his book *Quesiti et Inventioni Diverse* (*Various Tasks and Inventions*), published in 1546, Tartaglia answered questions about his origins and childhood in a dialogue: his father was a letter carrier on horseback and was called Michele. He does not know a family name. Asked why then he called himself *Tartaglia*, he recounted that in February 1512, when the French sacked his native city of Brescia and inflicted a terrible massacre, a soldier inflicted three wounds on his head and two on his face with sword blows, making him look like a monster if his full beard did not hide it. Among the injuries was one across the mouth and teeth, through which he could not speak properly for a time, but only stutter. Therefore, the children gave him the nickname *Tartaglia* (*stutterer*), which he kept as a souvenir of his misfortune. At that time he was about 12 years old. That is, he was born around the year 1500. In a document of 1529 a Nicolo from Brescia, master of arithmetic, certainly Tartaglia, is mentioned with an age of 30 years. This gives 1499 as the year of birth.

At the age of 14, as he further reported, Niccolò spent two weeks in a writing school learning the ABCs up to K. Then he ran out of money and stole a ready-made alphabet, with the help of which he taught himself the remaining letters. *…and so, from that day on, I was never again with any teacher, but only in the company of a daughter of poverty called Diligence*. In other words, all his knowledge of mathematics and military science he acquired as an autodidact in self-instruction. Tartaglia left Brescia around 1516, went via Crema, Bergamo and Milan to Verona, where he lived from about 1521 to 1534, and then moved to Venice, where he lived, with the exception of a year and a half stay in Brescia in 1548/49, until his death in 1557.

Tartaglia earned his living as a mostly commercial calculator and private tutor. Occasionally he gave lectures and, during the 18 months in Brescia, lectures on Euclid’s *Elements*, for which he received only a fraction of the fixed fee. A surviving list of his poor legacy shows the paucity in which one of the great mathematicians of the Italian Renaissance lived.

In February 1543, Tartaglia published the first translation of Euclid’s *Elements* into Italian, under the title *Euclide Megarense Philosopho: only introduction to the mathematical sciences… after the two translations*. The title is incorrect because Euclid of Megara was a philosopher who lived a century before the mathematician Euclid of Alexandria who is actually meant. The two translations used for this by Tartaglia, both Latin, were by Giovanni Campano, Latinized Johannes Campanus (1220-1296), printed in 1482, and by Bartolomeo Zamberti or Zamberto (1473-after 1543), printed in 1505. As a connoisseur of Euclid, Tartaglia was an expert on the fundamentals of geometry.

Tartaglia became famous not so much because of his books, but because he was involved in a heated dispute about the solution of cubic equations. Today one speaks of a single cubic equation x³ + ax² + bx + c = 0, where a, b and c can also be negative or 0, but at that time negative numbers were rejected. Therefore, 13 different cubic equations were distinguished: seven complete ones in which all powers are represented, three without a linear member and three without a quadratic member, namely in modern notation x³ + px = q, x³ = px + q and x³ + q = px. The third of these equations has a negative principal solution and was therefore usually not treated.

For a long time one had searched for a solution of the cubic equations. Finally, the lecturer of the University of Bologna Scipione dal Ferro (1465-1526) had found the solution of the first two equations without a quadratic member around 1505 or 1515, but had not published it. Such knowledge was, in fact, extremely valuable as an offensive or defensive weapon at a time when a university teacher’s reappointment and salary depended on how he performed in the frequent public scholarly contests in which the two opponents set each other tasks and problems.

Arithmetic masters also engaged in such mathematical battles, and so in early January 1535 Tartaglia and his Venetian rival Antonio Maria Fior also set each other 30 tasks to be solved within 40 or 50 days. Fior, as a student of dal Ferro, boasted of having the solution of the cubic equation (modern) x³ + px = q. All of Fior’s 30 problems were of this form. Thereupon, Tartaglia exerted himself and found the solution rule on February 12, 1535, and one day later also the one for the equation (modern) x³ = px + q. According to him, he solved all of Fior’s problems within two hours, while Fior could not solve a single one.

In the *Quesiti*, Tartaglia reports that on January 2, 1539, a bookseller from Milan appeared at his house. He had been sent by the physician Gerolamo Cardano (1501-1576),[3] who was considered a very great mathematician, was publicly reading Euclid in Milan, and was now having a work printed on the practice of arithmetic and geometry and on algebra. And because he had heard that Tartaglia, in a contest with Master Fior, had solved all 30 problems on the equation *Cosa e Cubo* (the unknown and the cube) equal to a number within two hours, “*he asks that you send him this rule you have discovered, and if it is agreeable to you, he will publish it in his present work under your name, and if it is not agreeable to you that he should publish it, he will keep it secret.*” Tartaglia’s reply, “*Tell his Excellency that you forgive me, but if I want to publish this invention of mine, it will be in my own works and not in those of others*.”

But Cardano did not let up. He pressed Tartaglia by letter and invited him to Milan under the pretext that the Spanish governor of Milan wanted to see him, and at Cardano’s house, according to Tartaglia, on March 25, 1539, the latter said, “*I swear to you by the Holy Gospels and as a true nobleman never to publish these discoveries of yours if you teach them to me*.” Tartaglia then told him the way to solve all three cubic equations in the form of a poem. And Tartaglia warned Cardano: “*If you do not keep the word of honor you have given me, I promise you to print a book immediately afterwards that will not be very pleasant to you*.”

Tartaglia could now have published his discovery. But he did not do so because he had no solution for the remaining ten cubic equations with a quadratic member, nor did he know what to do in the case of the (later called) casus irreducibilis, namely the case where square roots of negative numbers appear in the solution formula.

In 1539 and 1545 a book Cardano published under the title *Artis magnae sive de Regulis algebraicis Liber unus*, in which he published the solutions of cubic equations without a square member as the discovery of Scipione dal Ferros, but in two places he also gave Nicolaus Tartalea as the second discoverer. In this algebra book, Cardano showed how to transform cubic equations with a quadratic member into those with a linear member, and thereby lead them to a solution, which Tartaglia never succeeded in doing. That is, in this work one finds the instructions for solving all 13 cubic equations and also the 4th degree equations discovered by Cardano’s student Lodovico Ferrari (1522-1565).[4]

Tartaglia was seething with anger at Cardano’s betrayal. And he wrote the *Quesiti* in 1546 also to vilify Cardano in Task LX as doltish, endowed with little intelligence and reason, trembling in fear of a second-rate arithmetician, a poor sap and incapable of solving easy problems. Lodovico Ferrari then stepped up to defend his former teacher. On February 10, 1547, he addressed the first pamphlet (Italian: cartello) as a challenge to Tartaglia and sent it to numerous prominent Italian figures, whom he lists at the end of the twelve-page pamphlet. Ferrari, then 25 years old, challenged Tartaglia to a contest on geometry, arithmetic, and all disciplines dependent on them.

The two opponents exchanged six *cartelli* and six *risposte* (answers). The last one is dated July 24, 1548 by Tartaglia, who was already in Brescia at that time. In the second answer Tartaglia gives 31 tasks, in the third Cartello Ferrari as many. Both later declared that the opponent had not solved them or had not solved them correctly. Tartaglia taught Euclid in Brescia from March to the end of July 1548. When the hearers went to the country for the harvest, he decided to stop exchanging pamphlets with Ferrari and go to Milan for a public argument with Cardano and Ferrari. But Cardano, who had already stayed out of the discussion, left Milan, and so only Tartaglia and the brilliant mathematics lecturer Ferrari faced each other on August 10, 1548, in the church of Santa Maria del Giardino, located near the future Teatro alla Scala opera house. The majority of the audience was on Ferrari’s side, but this was not the only reason why Tartaglia lost out.

In May 1551, Tartaglia published a book of only 38 pages, the *General Rule for using reason and measure to lift not only any sunken ship but also a solid metal tower, called Travagliata Inventione (agonizing, laborious invention)*. At the same time, discussions of Nicolo Tartaglia about his* Travagliata Inventione*, a book of 48 pages, appeared. In the *Third Discussion* is told the reason to have titled his invention agonizing invention. “*I chose the title because I was under the greatest sufferings and agonies of my life when I found the main subject of this invention*” and then Tartaglia describes in 13 pages how he was cheated of his agreed payment during his Euclid lectures in Brescia in 1548/49.

In the last years of his life in Venice, Tartaglia wrote a great work on arithmetic, geometry and algebra, but only up to quadratic equations and without a word on cubic ones, the *General trattato di numeri et misure* (*General Treatise of Numbers and Measures*) in six parts, with many remarkable details – the best encyclopedia of mathematics of his time. The beginning appeared in 1556 while Tartaglia was still alive. The last parts came out posthumously in 1560.

‘

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can’t handle?, [13]

**References and Further Reading:**

- [1] Archimedes lifted the world off its Hinges, SciHi Blog
- [2] Euclid of Alexandria – the Father of Geometry, SciHi Blog
- [3] Gerolamo Cardano and Physician, Mathematician, and Gambler, SciHI Blog
- [4] Lodovico Ferrari and the quartic equations, SciHi Blog
- [5] Masotti, Arnoldo,
*Niccolò Tartaglia*in the*Dictionary of Scientific Biography*. - [6] Feldmann, Richard W. (1961). “The Cardano-Tartaglia dispute”.
*The Mathematics Teacher*.**54**(3): 160–163. - [7] Chisholm, Hugh, ed. (1911). .
*Encyclopædia Britannica*. Vol. 26 (11th ed.). Cambridge University Press. - [8] Herbermann, Charles, ed. (1913). .
*Catholic Encyclopedia*. New York: Robert Appleton Company. - [9] Tartaglia, Niccolò (1543).
*Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi*. Venice. - [10] Tartaglia, Niccolò (1543).
*Euclide Megarense philosopho*. Venice. - [11] Tartaglia, Niccolò (1556–1560),
*General Trattato di Numeri et Misure*, Venice: Curtio Troiano. - [12] O’Connor, John J.; Robertson, Edmund F., “Niccolò Fontana Tartaglia”,
*MacTutor History of Mathematics archive*, University of St Andrews - [13] 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can’t handle?, Mathologer @ youtube
- [14] Timeline of 16th Century Italian mathematicians, via DBpedia and Wikidata