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

On December 8, 1865, French mathematician **Jacques Salomon Hadamard** was born. Hadamard made major contributions in number theory, complex function theory, differential geometry and partial differential equations. Moreover, he is also known for his description of the mathematical though process in his book *Psychology of Invention in the Mathematical Field*.

“It is important for him who wants to discover not to confine himself to one chapter of science, but to keep in touch with various others.”

— Jacques Hadamard, [13]

Jacques Hadamard was born the son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, in Versailles, France. He attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. This was an unfortunate time for a child to be growing up in Paris, since the Franco-Prussian War which began on 19 July 1870 went badly for France and on 19 September 1870 the Prussians began a siege of Paris. This was a desperate time for the inhabitants, who were suffering from hunger. Paris surrendered on 28 January 1871 and the Treaty of Frankfurt, signed on 10 May 1871, was a humiliation for France. Between the surrender and the signing of the treaty there was essentially a civil war in Paris and the Hadamards‘ house was burnt down.[1]

In 1884 Hadamard entered the École Normale Supérieure, having been placed first in the entrance examinations both there and at the École Polytechnique. His teachers included amongst others mathematician and historian Paul Tannery, mathematician Charles Hermite [1], and mathematician Charles Émile Picard. While undertaking research for his doctorate he worked as a school teacher. Although his research went extremely well, his teaching was less appreciated, probably because he demanded more of his pupils than their abilities allowed. Hadamard obtained his doctorate in 1892 and in the same year was awarded the Grand Prix des Sciences Mathématiques for his essay on the Riemann zeta function.

In June of the same year, Hadamard married Louise-Anna Trénel, who had known each other from childhood. The couple moved to Bordeaux the following year when Hadamard was appointed as a lecturer at the University, where he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds. He published 29 papers during these four years, but they are remarkable more for their depth and the range of the topics which they covered rather than their number. In 1896 he made two important contributions: he proved the prime number theorem, using complex function theory (also proved independently by Charles Jean de la Vallée-Poussin); and he was awarded the Bordin Prize of the French Academy of Sciences for his work on geodesics in the differential geometry of surfaces and dynamical systems.

In the same year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on geometry and symbolic dynamics continued in 1898 with the study of geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.

After the Dreyfus affair [3], which involved him personally because his wife was related to Dreyfus, Hadamard became politically active and a staunch supporter of Jewish causes though he professed to be an atheist in his religion. Dreyfus was also of Jewish descent and embarked on a military career. Working at the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular, the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. At first Hadamard, like many people, assumed that Dreyfus was guilty. However after moving to Paris in 1897 he began to discover how evidence against Dreyfus had been forged.[1] Hadamard became a leading crusader to reopen the case against Dreyfus, who happened to be a relative of his wife. Eventually, Dreyfus was retried, found guilty again, and pardoned. Hadamard would not accept this and was among those who continued to press the government to clear Dreyfus’s name—a result finally achieved in 1906.[4]

In 1897 he moved back to Paris, holding positions in the Sorbonne and the Collège de France, where he was appointed Professor of Mechanics in 1909. In addition to this post, he was appointed to chairs of analysis at the École Polytechnique in 1912 and at the École Centrale in 1920. In Paris Hadamard concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the calculus of variations and the foundations of functional analysis. He introduced the idea of well-posed problem and the method of descent in the theory of partial differential equations, culminating in his seminal book on the subject, based on lectures given at Yale University in 1922.

“Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.”

– Jacques Hadamard, [13]

Among many other honors, Hadamard was elected to the French Academy of Sciences in 1916, in succession to Poincaré. At the beginning of the Second World War Hadamard escaped to southern France in 1940. The Vichy government permitted him to leave for the United States in 1941 and he obtained a visiting position at Columbia University in New York. In 1945 he published his reflections and investigations of the mathematical mind, entitled *The Psychology of Invention in the Mathematical Field*. Having lost his two older sons in World War I and another during World War II, he became active in international peace movements.[4] After the war, Hadamard returned to France, where he died in 1963, aged 97. Jacques Hadamard’s body of mathematical work is impressive both in its depth and in its breadth. In particular, his work transformed the theory of functions, contributed to the creation of functional analysis, and breathed new life into the theory of partial differential equations. Moreover, the influence of his legacy on the development of analysis in the 20th century and on the Bourbaki group is impossible to overstate.[6]

Ryan O’Donnell, *Hamming Code and Hadamard Code || @ CMU || Lecture 11c of CS Theory Toolkit*, [12]

**References and Further Reading:**

- [1] Jacques Hadamard at MacTutor History
- [2] Charles Hermite’s admiration for simple beauty in Mathematics, SciHi blog, Dec 24, 2014.
- [3] J’Accuse – Émile Zola and the Dreyfus Affaire, SciHi blog, January 13, 2013.
- [4] Jacques-Solomon Hadamard, at Britannica Online
- [5] S. Mandelbrojt: The Mathematical Work of Jacques Hadamard, in The American Mathematical Monthly Vol. 60, No. 9 (Nov., 1953), pp. 599-604.
- [6] Biography of Jacques Hadamard, at Fondation Hadamard
- [7] Works by or about Jacques Hadamard at Internet Archive
- [8] French Wikisource has original text related to this article: Jacques Hadamard
- [9] Jacques Hadamard at zbMATH
- [10] Jacques Hadamard at Mathematics Genealogy Project
- [11] Jacques Hadamard at Wikidata
- [12] Ryan O’Donnell,
*Hamming Code and Hadamard Code || @ CMU || Lecture 11c of CS Theory Toolkit*, Ryan O’Donnell @ youtube - [13] Jacques Hadamard,
*An Essay on the Psychology of Invention in the Mathematical Field*(1954) - [14] Timeline for Jacques Hadamard, via Wikisource

On December 1, 1947, English mathematician **G. H. Hardy** passed away. Hardy is known for his achievements in number theory and mathematical analysis, but also for his 1940 essay on the aesthetics of mathematics, A Mathematician’s Apology, and for mentoring the brilliant Indian mathematician Srinivasa Ramanujan.

“A mathematician … has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.”

— G. H. Hardy, in A Mathematician’s Apology (1940, 2012), 84.

Godfrey Harold Hardy was born on 7 February 1877, in Cranleigh, Surrey, England, the son of Isaac Hardy, a bursar and an art master at Cranleigh school, and his mother Sophia, who had been a senior mistress at Lincoln Training College for teachers. Both parents were mathematically inclined, but, coming from poor families, had not been able to have a university education.[1] Hardy’s own natural affinity for mathematics was perceptible already at an early age. When just two years old, he wrote numbers up to millions, and when taken to church he amused himself by factorising the numbers of the hymns. After schooling at Cranleigh, Hardy was awarded a scholarship to Winchester College for his mathematical work. In 1896 he entered Trinity College, Cambridge. After only two years of preparation, Hardy was fourth in the Mathematics Tripos examination, a result which continued to annoy him for, despite feeling that the system was very silly, he still felt that he should have come out on top.[1] Years later, he sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than a means to an end.

As the most important influence Hardy cites the self-study of* Cours d’analyse de l’École Polytechnique* by the French mathematician Camille Jordan, through which he became acquainted with the more precise mathematics tradition in continental Europe. In 1900 he passed part II of the tripos and was awarded a fellowship at Cambridge. In 1903 he earned his M.A., which was the highest academic degree at English universities at that time. From 1906 onward he held the position of a lecturer where teaching six hours per week left him time for research.

On the outbreak of World War I, Hardy volunteered for war service but was rejected on medical grounds. In 1919 he left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertrand Russell affair during World War I.[4] Hardy spent the academic year 1928–1929 at Princeton in an academic exchange with Oswald Veblen,[5] who spent the year at Oxford. Hardy left Oxford and returned to Cambridge in 1931, where he was Sadleirian Professor of Pure Mathematics until 1942. One reason for this was the higher reputation Cambridge enjoyed in mathematics, but also (according to C. P. Snow) that, unlike Oxford, he could continue to live at the college in retirement.

Hardy is credited with reforming British mathematics, introducing the rigorous conceptual clarification and proofs common in continental Europe. British mathematics had previously long drawn only on the reputation of Isaac Newton and was mainly concerned with applied problems. Hardy contrasted this with the tradition of the French *cours d’analyse* and aggressively advocated a “pure mathematics,” thus setting himself apart from the mathematical physics and applied mathematics (for example, hydrodynamics) practiced at Cambridge. From 1911 Hardy collaborated with John Edensor Littlewood, in extensive work in mathematical analysis and analytic number theory. This (along with much else) led to quantitative progress on the Waring’s problem, as part of the Hardy–Littlewood circle method, as it became known. In prime number theory, they proved results and some notable conditional results. This was a major factor in the development of number theory as a system of conjectures; examples are the first and second Hardy–Littlewood conjectures.

On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan’s manuscripts as a possible fraud. Hardy recognized some of Ramanujan’s formulae but others “*seemed scarcely possible to believe*“. After seeing Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy commented that “they [theorems] defeated me completely; I had never seen anything in the least like them before”. He figured that Ramanujan’s theorems “*must be true, because, if they were not true, no one would have the imagination to invent them*“. On 8 February 1913, Hardy wrote Ramanujan a letter expressing his interest in his work, adding that it was “*essential that I should see proofs of some of your assertions*“. In 1914, Ramanujan came to England and spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. Hardy was an atheist and an apostle of proof and mathematical rigor, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition.

Curiously, however, a relatively simple consideration in a letter to the editor of Science was enough to make him permanently famous among evolutionary biologists. He formulated this Hardy-Weinberg principle,[7] according to which the relative frequency of alleles in a gene pool remains constant over the generations, independently of Wilhelm Weinberg. Thus, of all people, the despiser of all applied mathematics became the founder of a branch of applied mathematics, population genetics. Apart from the Hardy-Weinberg principle, some of his results from number theory are now said to be used in cryptography.

Hardy preferred his work to be considered pure mathematics, perhaps because of his detestation of war and the military uses to which mathematics had been applied. Hardy deliberately pointed out in his apology that mathematicians generally do not “*glory in the uselessness of their work,*” but rather – because science can be used for evil as well as good ends – “*mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean*.”

Hardy’s book *A mathematicians apology* was written in 1940. It is one of the most vivid descriptions of how a mathematician thinks and the pleasure of mathematics.

“The mathematician’s pattern’s, like those of the painter’s or the poet’s, must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics.”

— G. H. Hardy, in A Mathematician’s Apology (1940, 2012).

One of Hardys quotes from C. P. Snow in the foreword to *A Mathematician´s Apology* is: “*Young men ought to be conceited, but they oughtn’t to be imbecile*“. Hardy referred here to a young man who praises *Finnegans Wake* by James Joyce as the greatest literary work of all time. Hardy reports similar experiences in an anecdote: One day he was sitting opposite a schoolboy on a train who was reading an elementary algebra schoolbook. Out of courtesy, he asked him after his reading, whereupon he received the condescending answer: “*It’s advanced algebra, you won’t understand*“. Already in his youth Hardy was an avowed atheist, later he went so far that he refused to enter the university chapel on formal occasions.

G. H. Hardy died on December 1, 1947, in Cambridge at age 70.

Raymond Flood, *Hardy, Littlewood, Ramanujan and Cartwright,* [12]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “G. H. Hardy“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] G. H. Hardy Biography, at The Famous People
- [3] 20th Century Mathematics: Hardy and Ramanujan, at The Story of Mathematics.
- [4] The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist, SciHi Blog
- [5] Oswald Veblen and Foundations of modern Topology, SciHi Blog
- [6] The Short Life of Srinivasa Ramanujan, SciHi Blog
- [7] Wilhelm Weinberg and the Genetic Equilibrium, SciHi Blog
- [8] Works by or about G. H. Hardy at Internet Archive
- [9] Hardy, G. H. (1999).
*Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work*. Providence, RI: AMS Chelsea. - [10] “G.H. Hardy”.
*Famous Mathematicians: Biography and Contributions of Great Mathematicians through History*. - [11] Titchmarsh, E.C. (1950). “Godfrey Harold Hardy”.
*J. London Math. Soc*.**25**(2): 81–138. - [12] Raymond Flood,
*Hardy, Littlewood, Ramanujan and Cartwright,*Gresham College @ youtube - [13] G. H. Hardy at Wikidata
- [14] G. H. Hardy at zbMATH
- [15] Timeline for G. H. Hardy, via Wikidata

On October 11, 1811, English mathematician, physicist, meteorologist, psychologist and pacifist **Lewis Fry Richardsen** was born. Richardson pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration.

Lewis Fry Richardson was the youngest of seven children of the tannery entrepreneur David Richardson and his wife Catherine, née Fry; the family had been Quakers since the 17th century. After attending elementary school in Newcastle, he went to the Bootham School in York from 1894, which had been founded in 1823 for sons of wealthy Quakers. From 1898 he attended Durham College of Science in Newcastle for two years, where he studied mathematics, physics, chemistry, botany and geology. He graduated with an Associate of Science, comparable to the A-levels of today. He then went on to study at King’s College, Cambridge, without specializing in any particular science. In 1903 he passed Part I of the Tripos in Natural Sciences.

Weather forecasting dates back to the Babylonians, predicting weather from cloud patterns and astrology around 650 BC. Aristotle wrote about weather patterns in Meteorologica and later on, Theophrastus compiled a book on weather forecasting, called the Book of Signs. In 904 AD, Ibn Wahshiyya’s Nabatean Agriculture discussed the weather forecasting of atmospheric changes and signs from the planetary astral alterations, signs of rain based on observation of the lunar phases, and weather forecasts based on the movement of winds.

The age of modern weather forecasting began around the time of the invention of the electric telegraph in 1835. The reason is probably that the telegraph technology allowed reports of weather conditions from a wide area to be received almost instantaneously. The field of scientifically forecasting weather was first researched by Royal Navy Officer Francis Beaufort [4] and his protégé Robert FitzRoy.[5] In 1859 the loss of the Royal Charter due to a storm inspired Robert FitzRoy to develop charts to allow predictions to be made, which he called “forecasting the weather“, thus coining the term “weather forecast“. Daily weather forecasting started in 1861 in *The Times*.

Sooner or later, it became necessary to develop a standard vocabulary describing the clouds. It was Luke Howard who, in 1802, classified and described clouds. Weather prediction was further enhanced with the increasing knowledge of atmospheric physics.

In 1922, Lewis Fry Richardson published his influencing work “*Weather Prediction By Numerical Process* “. Richardson described how small terms in the prognostic fluid dynamics equations governing atmospheric flow could be neglected, and a finite differencing scheme in time and space could be devised, to allow numerical prediction solutions to be found. Richardson further envisioned a large auditorium of thousands of people performing the calculations and passing them to others. However, the huge amount of calculations required was too large to be completed without the use of computers, and the size of the grid and time steps led to unrealistic results in deepening systems. In later research, it was found, through numerical analysis, that this was due to numerical instability. The first computerized weather forecast was performed by a team led by the mathematician John von Neumann [6] and resulted in the paper Numerical Integration of the Barotropic Vorticity Equation, published in 1950. Five years later, the practical use of numerical weather prediction began in 1955, spurred by the development of programmable electronic computers.

Richardson became a lecturer in physics and mathematics at Westminster Training College. During this time he began a second course of study in psychology and mathematics (B.Sc. in psychology and mathematics 1925). He skipped the M.Sc. because it was considered a venal degree and graduated with a D.Sc. in 1926. In the same year he was appointed a Fellow of the Royal Society (F.R.S.), the highest scientific honor in Britain. In that year, however, Richardson also decided to leave physical-meteorological research for good and devote himself to psychology. Here he worked in his scarce free time mainly on questions of psychophysics.

In 1929 he became director of the Technical College in Paisley, Scotland. The board of directors could hardly believe his luck that an F.R.S. was interested in such a position, which was connected with an immense teaching obligation. After the Nazis “seized power” in Germany, the Richardsons helped a number of German emigrants to gain a foothold in Britain. Disturbed by the political situation, Lewis Richardson increasingly shifted his activities to peace research. In August 1939, a few weeks before the beginning of World War II, he visited Gdansk to see for himself what the situation was like and traveled back via Berlin. In February 1940, Richardson resigned his position at the Technical College to devote himself entirely to peace research. The board of directors allowed the family to continue living in the director’s house; because Richardson was now living on his savings, the family had to cut back considerably.

Lewis Fry Richardson died on September 30, 1953, in Kilmun, Argyll and Bute, at age 71.

5 Things That Changed Weather Forecasting Forever, [9]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Lewis Fry Richardson”,
*MacTutor History of Mathematics archive*, University of St Andrews - [2] Lewis Fry Richardson at the European Geographysical Society
- [3] Lewis Fry Richardson at Britannica
- [4] Sir Francis Beaufort and the Wind Scale, SciHi Blog
- [5] Robert FitzRoy – From Darwin’s famous voyage to Meteorology, SciHi Blog
- [6] John von Neumann – Game Theory and the Digital Computer, SciHi Blog
- [7] Works by or about Lewis Fry Richardson at Internet Archive
- [8] Lewis Fry Richardson at Wikidata
- [9] 5 Things That Changed Weather Forecasting Forever, NASA’s Goddard Space Flight Center; John F. Kennedy footage courtesy of the John F. Kennedy Library Foundation, NASA Goddard @ youtube
- [10] Gold, E. (1954). “Lewis Fry Richardson. 1881-1953”.
*Obituary Notices of Fellows of the Royal Society*.**9**(1): 216–235. - [11] Peter Lynch (2008). “The origins of computer weather prediction and climate modeling”.
*Journal of Computational Physics*. University of Miami.**227**(7): 3436 - [12] Richardson, Lewis Fry (1922).
*Weather Prediction by Numerical Processes*. Boston: Cambridge University Press. - [13] Ashford, Oliver M. (1985).
*Prophet or Professor?: Life and Work of Lewis Fry Richardson*. Bristol: Adam Hilger. - [14] Hunt, J.C.R. (1998). “Lewis Fry Richardson and His Contributions to Mathematics, Meteorology, and Models of Conflict”.
*Annual Review of Fluid Mechanics*.**30**(1): xiii–xxxvi. - [15] Timeline for Lewis Fry Richardson, via Wikidata

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

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

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

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

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

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

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

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

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

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

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

Carl Sagan – Cosmos – Eratosthenes, [9]

**References and Further Reading:**

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

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

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

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

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

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

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

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

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

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

**References and Further Reading:**

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

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

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

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

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

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

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

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

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

**References and Further Reading:**

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