Sunday, 1 October 2023

# 1 from old math terms notes Obtuse, Amblygon

 Recovering some old notes I wrote for students over time and adding them as I go:

Obtuse is from the Latin formation ob (against) + tundere (to beat) and literally means to beat against. An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90 degrees but less than 180 degrees. A triangle with an obtuse angle is called an obtuse triangle.

You may (very rarely) encounter the name amblygon used for an obtuse triangle. It is also sometimes spelled ambligon. Amblygon is drawn from the Greek roots for blunt amblu preceding the root gon for angle (from knee). The use in English probably first occurred in Billingsley's translation of Euclid in 1570, although he wrote "amblygonum". Billingsley's translation was the first translation of Euclid's "Elements" in English. It was published at London in 1570 with the title The Elements of Geometric of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London

On This Day in Math - October 1


Scientists have one thing in common with children: curiosity. To be a good scientist you must have kept this trait of childhood, and perhaps it is not easy to retain just one trait. A scientist has to be curious like a child; perhaps one can understand that there are other childish features he hasn't grown out of.
~Otto Robert Frisch

The 274th day of the year; 274 is a tribonacci number..The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are 0, 0, 1, 1, 2, 4, 7,

274 is also the sum  of five cubes, 23 + 2 3 + 23 + 53 + 53, and of three triangular numbers 78 + 91 + 105. In 1796, Gauss proved that every positive integer could be expressed as the sum of (no more than) three triangular numbers.

274 is an example of Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity) 274= 2 * 137 and 2+ 7 +4 = 13 = 2 + 1 + 3 + 7. Find another. 


1386 University of Heidelberg founded. The Ruprecht-Karls-Universität Heidelberg (Heidelberg University, Ruperto Carola) is a public research university located in Heidelberg, Baden-Württemberg, Germany. It is the oldest university in Germany and was the fourth university established in the Holy Roman Empire. A coeducational institution since 1899, today Heidelberg consists of twelve faculties and offers degree programs at undergraduate, graduate and postdoctoral levels in some 100 disciplines. *Wik

1610 Lodovico Cigoli writes to Galileo to inform him that Father Christoph Clavius SJ, the senior mathematician at the Collegio Romano, had said that if the telescope revealed four
new ‘planets’ around Jupiter to Galileo, then Galileo must have put them in the telescope to begin with. Two months later, Clavius had observed Jupiter’s moons himself. *Albert Van Helden, Galileo and the telescope, The origins of the telescope, Royal Netherlands Academy of Arts and Sciences, Amsterdam 2010
The original Galileo telescope, which is preserved today at the Museo Galileo in Florence, Italy. (They wouldn't let me look through it.)

1648 In a letter to Samuel Hartlib, Sir Balthazar Gerbier sends a description of Pascal's mechanical calculator. Wikipedia describes Gerbier as "an Anglo-Dutch courtier, diplomat, art advisor, miniaturist and architectural designer." Nathan Kesling ‏@nathan13109

1658 The closing date for Pascal’s prize problems on the cycloid. (T. Christie advised me that some give the date as Oct 2).  A toothache earlier that year caused him to return to mathematics and to study the cycloid. In 1654, late in the evening Pascal experienced a religious ecstasy that called him to give up his intermittent interest in mathematics and to devote his time to religious contemplation. For years he devoted no time to mathematics. Then one night, unable to sleep because of an abscessed tooth, Pascal began to think about some problems about the cycloid. His pain disappeared and he interpreted this as a sign that God was pleased by his mathematical studies. In a brief time he completed the investigation of the cycloid. Then he established a contest about the cycloid with himself and Roberval as the judges. The three problems he asked were:

1. Find the area and the center of gravity of the region BCD bounded by the cycloid, the horizontal line BC and the axis of symmetry AD.
2. Find the volume and center of gravity of the solids obtained by revolving the region BCD about AD and about BC.
3. For the solids in the previous question, find the center of gravity of the solids formed when each is cut by a plane parallel to its axis of revolution.
Only two contestants submitted solutions. No prize was awarded as the judges declared that the solutions were either incomplete or incorrect. Pascal then published his own results in a paper entitled "L'Histoire de la Roulette".  It is worth noting that all these investigations of the cycloid occurred before Newton and Leibnitz' work on the calculus!  *Historical Modules for the Mathematical Classroom There is a famous statue by Pajou in the Louvre of Pascal in which he is contemplating the Roulette (cycloid). (And in this photo, I am contemplating him contemplating the roulette) More on the cycloid, including a close up of the tablet in the statue is here.

1670 James Gregory writes to John Collins, with the first use of what will come to be called the Newton-Gregory interpolation formula. He includes in the letter two enclosures showing how to apply his method to series for sines and logarithms. *Beery & Stedall, Thomas Harriot’s Doctrine of Triangular Numbers, pg 51-52

1752  A letter of Benjamin Franklin written on October 1st, to Mr. Peter Collinson, FRS concerning an electrical kite, was read before the society on Dec 21.  Franklin describes the construction of the kite from two light strips of cedar and a large thin silk  handkerchief.
Benjamin Franklin Drawing Electricity from the Sky, an artistic rendition of Franklin's kite experiment painted by Benjamin West c. 1816

1812   Thomas Jefferson writes to William Duanne.  "The hand of age is upon me....last year it was the sight, this it is the hearing... but the mind is too weakened, When I was young, mathematics were the passion of my life. The same passion has returned upon me... 

1814 The terms "commutative" and "distributive" were used (in French) by François Joseph Servois in a memoir published in Annales de Gergonne (volume V, no. IV) *Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

1831 Michael Faraday discovers induced electric current using a helix made of two coils each of 203 feet of insulated copper wire. "A sudden jerk was perceived when the battery communication was made and broken... it was one way when made, and the other when broken." *A history of physics in its elementary branches By Florian Cajori

1831 (exact date unkonwn, October ?) Galois, while in prison, writes a preface to his "Memoires". As late as 1906, it had never been published, and Jules Tannery, reviewing the yet unpublished works of Galois, left it out, regarding it as too much like drunken raving. The ending is a fascinating hope for the future of mathematical publishing:
Evariste Galois 1811–1832
By Laura Toti Rigatelli

1842 Arthur Cayley's acceptance to Trinity was announced on this day. He was twenty-one years old and accepted on his first sitting, a rare event. He was the youngest man admitted to Trinity in the 19th Century. * A. J. Crilly, Arthur Cayley: Mathematician Laureate of the Victorian Age

1847 Maria Mitchell sees a comet... the first woman astronomer in the United States discovered a comet. On this night in the Autumn of 1847, Maria looked at the sky through the telescope in her homemade observatory at Nantucket, Mass. and saw a star five degrees above the North Star where there had been no star before. She had memorized the sky and was sure of her observation. It occurred to her that this might be a comet. Maria recorded the presumed comet's coordinates. The next night the star moved again. This time she was sure it was a comet. For this discovery, she was awar *New England Historical Sded a gold medal by the king of Denmark. She became the first woman elected to the American Academy of Arts and Sciences. *TIS
Maria Mitchell with student at Vassar Observatory  *New England Historical Society

1861 On Oct 1, a seemingly depressed Charles Darwin writes, "My Dear Lyell, ... I am very poorly today & very stupid & hate everybody & everything. One lives only to make blunders.–... I am
Ever yours
C. Darwin

1891 On Oct 1 Stanford University​ opened its doors after six years of planning and building. The prediction of a New York newspaper that Stanford professors would "lecture in marble halls to empty benches" was quickly disproved. The first student body consisted of 555 men and women, and the original faculty of 15 was expanded to 49 for the second year. The university’s first president was David Starr Jordan​, a graduate of Cornell, who left his post as president of Indiana University​ to join the adventure out West.
The Stanfords engaged Frederick Law Olmsted​, the famed landscape architect who created New York’s Central Park​, to design the physical plan for the university. The collaboration was contentious, but finally resulted in an organization of quadrangles on an east-west axis. Today, as Stanford continues to expand, the university’s architects attempt to respect those original university plans. *Stanford Univ Web page

1895  On the first of October 1895, the first German institute of insurance science was founded at the University of Göttingen, as a result of joint efforts of Felix Klein (1849-1825) and his fellow student Ludwig Kiepert (1846-1934), who was then chairman of the Prussian Civil Service Association (today called Hannover Life Insurance).This was the first institute in Germany in which a curriculum in actuarial mathematics, insurance law, and insurance economics was offered. Successful studies led to the degree “Versicherungsverständiger” (insurance expert). The institute was divided into a mathematical section and an administrative section, and its first chairman was Wilhelm Lexis. *From Center for Statistics, History of Statistics in Gottingen.

1907 Delegates from 310 Esperanto societies throughout the world met to elect a committee to modify the language. Louis Couturat, influenced by Leibniz’s thought on the construction of a logical universal language, was elected one of the secretaries.  *VFR

1934 Paul Erdos stops in Cambridge to visit with mathematical friends, particularly Harrold Davenport and Richard Rado, on his way to a position in Manchester. *Bruce Schechter, My Brain is Open: The Mathematical Journeys of Paul Erdos

1954 IBM announced is 705 EDP, part of its 700 series of mainframe computers. A business-oriented machine, the 705 had magnetic core memory.*CHM

1957 Thalidomide was first marketed on this date. This notorious drug was marketed as a mild sleeping pill that was safe even for pregnant women. It wasn’t until 1962 that the severe side effects were revealed, where it had caused the development of malformed limbs in babies. *

1958  On July 29, 1958, Eisenhower signed the National Aeronautics and Space Act, establishing NASA. When it began operations on October 1, 1958, NASA absorbed the 43-year-old NACA intact; its 8,000 employees, an annual budget of US$100 million, three major research laboratories (Langley Aeronautical Laboratory, Ames Aeronautical Laboratory, and Lewis Flight Propulsion Laboratory) and two small test facilities  *Wik
1969 Concorde goes Mach 1 In 1969, the prototype French-built Concorde broke the sound barrier for the first time. The inaugural flight of the aircraft had taken place on 2 Mar 1969 in Toulouse, France, and its first commercial flight was on 21 Jan 1976. It was the first plane in the world to be entirely controlled by computer. As the only supersonic passenger aircraft, the Anglo-French Concorde remains a brilliant technological achievement, though its impact on international air travel has been limited by the high cost of buying and operating the aircraft. There was also widespread opposition from environmental groups on the grounds of the Concorde's noise on takeoff and its fuel consumption. Only British Airways and Air France have operated the aircraft. *TIS

1971 The first CT (Computed Tomography (CT) imaging is also known as "CAT scanning" (Computed Axial Tomography)Tomography is from the Greek word "tomos" meaning "slice" or "section" and "graphia" meaning "describing") scan of a patient was performed OTD in 1971 at the Atkinson Morley Hospital in Wimbledon and highlighted a brain cyst.  The tomograph was built by British engineer Godfrey Hounsefield that year. *StoriaMedicina

1988   The game Connect Four Solved first by James D. Allen (Oct 1, 1988), and independently by Victor Allis (Oct 16, 1988). First player can force a win. Strongly solved by John Tromp's 8-ply database (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (Feb 18, 2006). *Wik

2008  A Metrolink engineer at the helm of a commuter train in Los Angeles, California, was found to have been text messaging seconds before colliding with a freight train. 25 people were killed in the accident and numerous others were injured. Many states have passed laws enforcing hands-free only cellular use that restricts drivers from the distracted driving inherent in hands-on texting and cellular phone calls. *CHM

2012 With God's grace, Dame Kathleen Ollerenshaw will awake for her 100th birthday today. Happy Birthday to a Grand-Ol-Dame, and may a puzzle occupy her thoughts. (See 1912 Births below).
(she did indeed greet her 100 th birthday, but Died: August 10, 2014, Didsbury, Manchester, United Kingdom)


1535 Giambattista della Porta (? Oct 1535 - 4 Feb 1615) Italian natural philosopher, experimenter and mathematician, though he also sought the miraculous or magical. He studied optics, including refraction (De refractione, 1593). Porta did not invent the telescope, regardless of his published claim. He was the first to propose adding a convex lens to the camera obscura, and first to recognize the heating effect of light rays. He wrote on cryptography in De furtivis literarum (1563), and his other books included mechanics, squaring the circle, description of a steam engine in De spiritali (1606). He formed the society, Accademia dei Segreti, dedicated to discussing and studying nature, meeting at his home, until closed by the Inquisition (about 1578). *TIS

1671 Luigi Guido Grandi was an Italian Jesuit who worked on geometry and hydraulics.Grandi was the author of a number of works on geometry in which he considered the analogies of the circle and equilateral hyperbola. He also considered curves of double curvature on the sphere and the quadrature of parts of a spherical surface.
In 1701 Grandi discussed the conical loxodrome, the curve that cuts the generators of a cone of revolution in a constant angle. He studied the curve the Witch of Agnesi in 1703. In fact his work of 1703 is important in introducing Leibniz's calculus into Italy.
In 1728 Grandi published Flores geometrici a work in which he defines the clelie curve. He named the curve after Countess Clelia Borromeo and dedicated his book to her. If the longitude and colatitude of a point P on a sphere is denoted by θ and φ and if P moves so that θ = m φ, where m is a constant, then the locus of P is a clelie. Grandi also applied the term "clelies" to the curves determined by certain trigonometric equations involving the sine function
a sin θ = b sin mφ
a sin θ = a - b sin mφ
Grandi also worked on hydraulics and was involved with a number of projects such as ones to drain the Chiana Valley and the Pontine Marshes. He also published a number of works on mechanics and astronomy. His practical work on mechanics included experimenting with a steam engine. *SAU 
He is noted for the roses that he introduced. His idea was to find a geometrical definition of curves which resemble flowers. These curves are still part of our calculus courses, except now we use polar coordinates to define them.*VFR

1873 Alfreds Arnolds Adolfs Meders (1 Oct 1873 , 1944) Meders worked on differential geometry and mathematical analysis. He often published papers written in German, in German journals. For example he published the following three papers in Crelle's Journal: Über einige Arten Singularer Punkte von Raumkurven (1896); Zur Theorie der singularen Punkte einer Raumkurve (1899); and Analytische Untersuchung singularer Punkte von Raumkurven (1910). In Monatshefte für Mathematik he published: Über die Determinante von Wronski (1906); and Zur Differentiation bestimmter Integrale nach einem Parameter (1911).
Meders was also interested in the history of mathematics and he wrote an important paper Direkte und indirekte Beziehungen zwischen Gauss und der Dorpater Universität (Direct and indirect connections between Gauss and the University of Dorpat) in 1928. His interests went outside mathematics and he sometimes lectured on astronomy, meteorology and biology where he had a special interest in birds. *SAU

1898 Béla Kerékjártó (October 1, 1898, –June 26, 1946) was a Hungarian mathematician who wrote numerous articles on Topology. He earned his Ph.D. degree from the University of Budapest. He taught at the Faculty of Sciences of the University of Szeged from 1922, and at the University of Budapest from 1938. In 1923, he published one of the first books on Topology; Hermann Weyl wrote that this book completely changed his views of the subject.*Wik

1904 Otto Robert Frisch (1 Oct 1904; 22 Sep 1979) Austrian-British nuclear physicist, born in Vienna, who, with his aunt Lise Meitner, described the division of neutron-bombarded uranium into lighter elements. He named the process fission, borrowing a term from biology (1939). At the time, Meitner was working in Stockholm and Frisch (1934-39) at Copenhagen under Niels Bohr, who brought their observation to the attention of Albert Einstein and others in the United States. He did research with James Chadwick 1940-43, and was head of the Critical Assembly Group on the Los Alamos project 1943-46. After World War II, Frisch became a science writer on atomic physics for the layman. *TIS

1911 Zhou Weiliang (simplified Chinese: (October 1, 1911– August 10, 1995) was a Chinese mathematician born in Shanghai, known for his work in algebraic geometry.
He was a student in the USA, graduating from the University of Chicago in 1931. In 1932 he attended the University of Göttingen, then transferring to Leipzig where he worked with van der Waerden. They produced a series of joint papers on intersection theory, introducing in particular the use of what are now generally called Chow coordinates (which were in some form familiar to Arthur Cayley).
He married Margot Victor in 1936, and took a position at the National Central University in Nanjing. His mathematical work was seriously affected by the wartime situation in China. He taught at the National Tung-Chi University in Shanghai in the academic year 1946–47, and then went to the Institute for Advanced Study in Princeton, where he returned to his research. From 1948 to 1977 he was a professor at Johns Hopkins University. *Wik

1912 Dame Kathleen Mary Ollerenshaw, née Timpson, DBE (1 October 1912, August 10, 2014, Didsbury, Manchester, United Kingdom ) is a British mathematician and politician. Deaf since the age of eight, she loved doing arithmetic problems as a child. As a young woman, she attended St Leonards School and Sixth Form College in St Andrews, Scotland where today the house of young male boarders is named after her. At the age of 19, she gained admittance to Somerville College, Oxford to study mathematics. She completed her doctorate at Somerville in 1945 on "Critical Lattices" under the supervision of Theo Chaundy. She wrote five original research papers which were sufficient for her to earn her DPhil degree without the need of a formal written thesis.
Ollerenshaw served as a Conservative Councillor for Rusholme for twenty-six years (1956–1981), was Lord Mayor of Manchester (1975–1976), and the prime motivator in the creation of the Royal Northern College of Music. She was made a Freeman of the City of Manchester and was an advisor on educational matters to Margaret Thatcher's government in the 1980s.
She has published at least 26 mathematical papers, her best-known contribution being to most-perfect pandiagonal magic squares. An annual public lecture at the School of Mathematics, University of Manchester is named in her honour.
An amateur astronomer, Ollerenshaw donated her telescope to Lancaster University, and an observatory there bears her name. She is an honorary member of the Manchester Astronomical Society and held the post of Vice President for a number of years. *Wik A wonderful article about her approaching her 100th birthday is in Scientific American.


1768 Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The pedal line of a triangle is sometimes called the "Simson line" after him. Edmond Halley suggested to him that he might devote his considerable talents to the restoration of the work of the early Greek geometers, such as Euclid and Apollonius of Perga. These are works that only survive in abbreviated accounts given by later mathematicians such as Pappus of Alexandria. He first studied Euclid's so-called porisms. Playfair's 1792 definition of porism is "a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions."
Simson's work on Euclid's porisms was published in 1723 in the Philosophical Transactions of the Royal Society, and his restoration of the Loci Plani of Apollonius appeared in 1749. Further work of his on porisms and other subjects including logarithms was published posthumously in 1776 by Lord Stanhope at his own expense. Simson also set himself the task of preparing an edition of Euclid's Elements in as perfect a form as possible, and his edition of Euclid's books 1-6, 11 and 12 was for many years the standard text and formed the basis of textbooks on geometry written by other authors. The work ran through more than 70 different editions, revisions or translations published first in Glasgow in 1756, with others appearing in Glasgow, Edinburgh, Dublin, London, Cambridge, Paris and a number of other European and American cities. Recent editions appeared in London and Toronto in 1933 under the editorship of Isaac Todhunter, and in São Paolo in 1944. Simson's lectures were delivered in Latin, at any rate at the beginning of his career. His most important writings were written in that language, however, his edition of Euclid, after its first publication in Latin, appeared in English, as did a treatise on conic sections that he wrote for the benefit of his students.
the Simson line does not appear in his work but Poncelet in Propriétés Projectives says that the theorem was attributed to Simson by Servois in the Gergonne's Journal. It appears that the theorem is due to William Wallace.
The University of St Andrews awarded Simson an honorary Doctorate of Medicine in 1746.
In 1753 Simson noted that, as the Fibonacci numbers increased in magnitude, the ratio between adjacent numbers approached the golden ratio, whose value is
(1 + √5)/2 = 1.6180 . . . . *SAU

1924 John Edward Campbell is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU

1972 Francisco José Duarte (6 Jan 1883, 1 Oct 1972) Duarte's most important work in mathematics was done in algebra, number theory and mathematical analysis. His first work in mathematics was about which he presented to the Paris Academy of Sciences in 1907. He published papers on the general solution of a diophantine equation of the third degree x3 + y3 + z3 - 3xyz = v3, simplified Kummer's criterion and gave a simple proof of the impossibility of solving the Fermat equation x3 + y3 + z3 = 0 in nonzero integers. He also observed that the interpolation formula of Everett is a consequence of the interpolation formula of Gauss. In 1908 he published an article where he calculated π to 200 decimal places.
His main three books are: Monograph on the numbers π and e. Historical and bibliographical notes (Spanish) (Bol. Acad. Cien. Fis. Mat. Nat. 11(1948)), with 27 chapters on 250 pages, which contains more information on π and e than has ever before been collected in one place; Lessons on Infinitesimal Analysis (Caracas 1943, 606 pp.) (Spanish) containing material from courses in analysis at UCV during his first three or four years there; and Bibliography of Euclid, Archimedes, Newton (Acad. Cien. Fis. Mat. Nat., Caracas 1963, 163 pp.) (Spanish) which was also done in the 19th century.
Many mathematicians are interested in recreational mathematics. Duarte also contributed to that part of mathematics and proposed problems and solutions to the American Mathematical Monthly for several years, and also to the journal Ciencia y Ingenieria (Science and Engineering) published in Mérida. *SAU

1990 John Stewart Bell​ FRS (28 June 1928 – 1 October 1990) was a physicist from Northern Ireland (Ulster), and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.*Wik

1996 Herbert Karl Johannes Seifert (May 27, 1907– October 1, 1996) was a German mathematician known for his work in topology.

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Saturday, 30 September 2023

On This Day in Math - September 30

Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
~Lewis Richardson

The 273rd day of the year; 273oK(to the nearest integer)is the freezing point of water, or 0oC

OOOOH wait, 273 = 13*7*3, and 1373 is also prime.. and There are only two sphenic numbers consisting of concatenation of distinct prime numbers, this is the smaller of the two.(sphenic or wedge numbers are products of three distinct primes) *Prime curios

273 is a repdigit in hexdecimal, or base 16 (111) 16^2 + 16 + 1, and in base 20(vigesimal), where it looks like a bad report card (DD)

273 = 47^2 - 44^2 and also 137^2 - 136^2

Prime Curios says that 273^10 - 10^273 is the smallest n for which this expression is prime. There are no more year days that exhibit this relationship.

Prime Curios includes this tasty factoid, the prime factors of 273 = 3 x 7 x 13, and if concatenated in reverse order, 1373, it is prime.

273 is a Palindrome in base 2(100010001). 2^8 + 2^4 + 2^0. Also a Palindrome in base 4 (10101) , 4^4 + 4^2 + 4^0.

273 is the product of two consecutive Fibonacci numbers (13 x 21) and is thus a Golden rectangle Number. Each of these have side ratios that approximate a Golden Rectangle as they grow larger . 21/13=1.615....

There are 273 distinct ways to connect 6 points with 5 straight non-crossing lines. Here they are:

Only 28 are unique up to reflection and rotation,


1717 Colin Maclaurin (1698–1746), age 19, was appointed to the Mathematics Chair at Marischal College, Aberdeen, Scotland. This is the youngest at which anyone has been elected chair (full professor) at a university. (Guinness) In 1725 he was made Professor at Edinburgh University on the recommendation of Newton. *VFR

At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending a thesis on the Power of Gravity, and remained at Glasgow to study divinity until he was 19, when he was elected professor of mathematics in a ten-day competition at Marischal College and University in Aberdeen. This record as the world's youngest professor endured until March 2008, when the record was officially given to Alia Sabur, Alia Sabur.  On 19 February 2008, at 18 years of age (3 days before her 19th birthday), she was appointed to the position of International Professor as Research Liaison with Stony Brook University by the Dept. of Advanced Technology Fusion at Konkuk University in Seoul, South Korea. The position was a temporary, one-year contract which she chose not to renew.

Colin MacLaurin *Wik

1810 The University of Berlin opened. *VFR It is now called The Humboldt University of Berlin and is Berlin's oldest university. It was founded as the University of Berlin (Universität zu Berlin) by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities.*Wik

1846 First use of either for tooth extraction by American dentist William T G Morten.  He would give a public demonstration on October 16 of his use of the anesthesia. 

*Lucio Gelmini@gelminil7

1890 In his desk notes Sir George Biddell Airy writes about his disappointment on finding an error in his calculations of the moon’s motion. “ I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.” *George Biddell Airy and Wilfrid Airy (ed.), Autobiography of Sir George Biddell Airy (1896), 350.

1893 Felix Klein visits Worlds fair in Chicago, then visits many colleges. On this day the New York Mathematical society had a special meeting to honor him. *VFR

1921 William H Schott patented the "hit-and-miss synchronizer for his clocks. The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shott in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK. They were the most accurate pendulum clocks ever commercially produced, and became the highest standard for timekeeping between the 1920s and the 1940s, after which mechanical clocks were superseded by quartz time standards. They were used worldwide in astronomical observatories, naval observatories, in scientific research, and as a primary standard for national time dissemination services. The Shortt was the first clock to be a more accurate timekeeper than the Earth itself; it was used in 1926 to detect tiny seasonal changes (nutation) in the Earth's rotation rate. *Wik


 The first manned rocket-powered flight was made by German auto maker Fritz von Opel. His Sander RAK 1 was a glider powered by sixteen 50 pound thrust rockets. In it, Opel made a successful flight of 75 seconds, covering almost 2 miles near Frankfurt-am-Main, Germany. This was his final foray as a rocket pioneer, having begun by making several test runs (some in secret) of rocket propelled vehicles. He reached a speed of 238 km/h (148 mph) on the Avus track in Berlin on 23 May, 1928, with the RAK 2. Subsequently, riding the RAK 3 on rails, he pushed the world speed record up to 254 km/h (158 mph). The first glider pilot to fly under rocket power, was another German, Friedrich Staner, who flew about 3/4-mile on 11 Jun 1928.*TIS

1938  On this day in 1938, Olga Taussky married Jack Todd. 50 years later she said:-
"My life and my career would have been so different if my Irishman had not come along."
Olga Taussky-Todd was an Austrian born mathematician who worked on algebraic number theory and matrix theory.
John Todd was an Irish-born numerical analyst. 

Olga Taussky photo by Paul Erdos

2010 The ignoble prizes, presented on this date, included an engineering for collecting Whale Snot, and a MANAGEMENT PRIZE: for demonstrating mathematically that organizations would become more efficient if they promoted people at random. See all the 2010 winners here.


1550 Michael Maestlin (30 September 1550, Göppingen – 20 October 1631, Tübingen) was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centerd orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU The first known calculation of the reciprocal of the golden ratio as a decimal of "about 0.6180340" was written in 1597 by Maestlin in a letter to Kepler. He is also remembered for :
Catalogued the Pleiades cluster on 24 December 1579. Eleven stars in the cluster were recorded by Maestlin, and possibly as many as fourteen were observed.
Occultation of Mars by Venus on 13 October 1590, seen by Maestlin at Heidelberg. *Wik

1715 Étienne Bonnot de Condillac (30 Sep 1715; 3 Aug 1780) French philosopher, psychologist, logician, economist, and the leading advocate in France of the ideas of John Locke (1632-1704). In his works La Logique (1780) and La Langue des calculs (1798), Condillac emphasized the importance of language in logical reasoning, stressing the need for a scientifically designed language and for mathematical calculation as its basis. He combined elements of Locke's theory of knowledge with the scientific methodology of Newton; all knowledge springs from the senses and association of ideas. Condillac devoted careful attention to questions surrounding the origins and nature of language, and enhanced contemporary awareness of the importance of the use of language as a scientific instrument.*TIS

1774 Carl Wilhelm Scheele sent a letter to Antoine Lavoisier announcing the discovery of oxygen (O). Unfortunately the letter from the Swedish chemist was never acknowledged and Joseph Priestly published the discovery first. Scheele was trounced in the announcement of other discoveries as well, he identified molybdenum, tungsten, barium, hydrogen, and chlorine before Humphry Davy, among others. Scheele discovered organic acids tartaric, oxalic, uric, lactic, and citric, as well as hydrofluoric, hydrocyanic, and arsenic acids. (Not bad for a chemist you never heard of.) For this reason, Isaac Asimov nicknamed him “hard-luck Scheele” * , *Wik
An interpretation of Scheele from the late 19th or early 20th century as no contemporary portraits of him are known (by xylographer Ida Amanda Maria Falander (1842-1927))

1775 Robert Adrain (30 September 1775 – 10 August 1843) . Although born in Ireland he was one of the first creative mathematicians to work in America. *VFR Adrain was appointed as a master at Princeton Academy and remained there until 1800 when the family moved to York in Pennsylvania. In York Adrain became Principal of York County Academy. When the first mathematics journal, the Mathematical Correspondent, began publishing in 1804 under the editorship of George Baron, Adrain became one of its main contributors. One year later, in 1805, he moved again this time to Reading, also in Pennsylvania, where he was appointed Principal of the Academy.
After arriving in Reading, Adrain continued to publish in the Mathematical Correspondent and, in 1807, he became editor of the journal. One has to understand that publishing a mathematics journal in the United States at this time was not an easy task since there were only two mathematicians capable of work of international standing in the whole country, namely Adrain and Nathaniel Bowditch. Despite these problems, Adrain decided to try publishing his own mathematics journal after he had edited only one volume of the Mathematical Correspondent and, in 1808, he began editing his journal the Analyst or Mathematical Museum.
With so few creative mathematicians in the United States the journal had little chance of success and indeed it ceased publication after only one year. After the journal ceased publication, Adrain was appointed professor of mathematics at Queen's College (now Rutgers University) New Brunswick where he worked from 1809 to 1813. Despite Queen's College trying its best to keep him there, Adrain moved to Columbia College in New York in 1813. He tried to restart his mathematical journal the Analyst in 1814 but only one part appeared. In 1825, while he was still on the staff at Columbia College, Adrain made another attempt at publishing a mathematical journal. Realising that the Analyst had been too high powered for the mathematicians of the United States, he published the Mathematical Diary in 1825. This was a lower level publication which continued under the editorship of James Ryan when Adrain left Columbia College in 1826. *SAU
portrait by Charles C. Ingham *Wik

1870 Jean-Baptiste Perrin (30 Sep 1870; 17 Apr 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein's explanation of this phenomenon and thereby confirmed the atomic nature of matter. Using a gamboge emulsion, Perrin was able to determine by a new method, one of the most important physical constants, Avogadro's number (the number of molecules of a substance in so many grams as indicated by the molecular weight, for example, the number of molecules in two grams of hydrogen). The value obtained corresponded, within the limits of error, to that given by the kinetic theory of gases. For this achievement he was honoured with the Nobel Prize for Physics in 1926.*TIS

1882 Hans Wilhelm Geiger  (30 Sep 1882; 24 Sep 1945) was a German physicist who introduced the Geiger counter, the first successful detector of individual alpha particles and other ionizing radiations. After earning his Ph.D. at the University of Erlangen in 1906, he collaborated at the University of Manchester with Ernest Rutherford. He used the first version of his particle counter, and other detectors, in experiments that led to the identification of the alpha particle as the nucleus of the helium atom and to Rutherford's statement (1912) that the nucleus occupies a very small volume in the atom. The Geiger-Müller counter (developed with Walther Müller) had improved durability, performance and sensitivity to detect not only alpha particles but also beta particles (electrons) and ionizing electromagnetic photons. Geiger returned to Germany in 1912 and continued to investigate cosmic rays, artificial radioactivity, and nuclear fission.*TIS
A "two-piece" bench-type Geiger–Müller counter using a cylindrical end-window detector connected to an electronics module with analogue readout

1883 Ernst David Hellinger (1883 - 1950) introduced a new type of integral: the Hellinger integral . Jointly with Hilbert he produced an important theory of forms. *SAU

1894 Dirk Jan Struik (30 Sept 1894 , 21 Oct 2000) Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.
In 1924, funded by a Rockefeller fellowship, Struik traveled to Rome to collaborate with the Italian mathematician Tullio Levi-Civita. It was in Rome that Struik first developed a keen interest in the history of mathematics. In 1925, thanks to an extension of his fellowship, Struik went to Göttingen to work with Richard Courant compiling Felix Klein's lectures on the history of 19th-century mathematics. He also started researching Renaissance mathematics at this time.
Struik was a steadfast Marxist. Having joined the Communist Party of the Netherlands in 1919, he remained a Party member his entire life. When asked, upon the occasion of his 100th birthday, how he managed to pen peer-reviewed journal articles at such an advanced age, Struik replied blithely that he had the "3Ms" a man needs to sustain himself: Marriage (his wife, Saly Ruth Ramler, was not alive when he turned one hundred in 1994), 
Mathematics, and Marxism.
It is therefore not surprising that Dirk suffered persecution during the McCarthyite era. He was accused of being a Soviet spy, a charge he vehemently denied. Invoking the First and Fifth Amendments of the U.S. Constitution, he refused to answer any of the 200 questions put forward to him during the HUAC hearing. He was suspended from teaching for five years (with full salary) by MIT in the 1950s. Struik was re-instated in 1956. He retired from MIT in 1960 as Professor Emeritus of Mathematics.
Aside from purely academic work, Struik also helped found the Journal of Science and Society, a Marxian journal on the history, sociology and development of science.
In 1950 Stuik published his Lectures on Classical Differential Geometry.
Struik's other major works include such classics as A Concise History of Mathematics, Yankee Science in the Making, The Birth of the Communist Manifesto, and A Source Book in Mathematics, 1200-1800, all of which are considered standard textbooks or references.
Struik died October 21, 2000, 21 days after celebrating his 106th birthday. *Wik

1905 Sir Nevill F. Mott (30 Sep 1905; 8 Aug 1996) English physicist who shared (with P.W. Anderson and J.H. Van Vleck of the U.S.) the 1977 Nobel Prize for Physics for his independent researches on the magnetic and electrical properties of amorphous semiconductors. Whereas the electric properties of crystals are described by the Band Theory - which compares the conductivity of metals, semiconductors, and insulators - a famous exception is provided by nickel oxide. According to band theory, nickel oxide ought to be a metallic conductor but in reality is an insulator. Mott refined the theory to include electron-electron interaction and explained so-called Mott transitions, by which some metals become insulators as the electron density decreases by separating the atoms from each other in some convenient way.*TIS

1913 Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish and American mathematician born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg–Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974. *Wik

1916 Richard Kenneth Guy (born September 30, 1916, Nuneaton, Warwickshire -  March 9,  2020 ) is a British mathematician, and Professor Emeritus in the Department of Mathematics at the University of Calgary.
He is best known for co-authorship (with John Conway and Elwyn Berlekamp) of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory, but he has also published over 100 papers and books covering combinatorial game theory, number theory and graph theory.
He is said to have developed the partially tongue-in-cheek "Strong Law of Small Numbers," which says there are not enough small integers available for the many tasks assigned to them — thus explaining many coincidences and patterns found among numerous cultures.
Additionally, around 1959, Guy discovered a unistable polyhedron having only 19 faces; no such construct with fewer faces has yet been found. Guy also discovered the glider in Conway's Game of Life.
Guy is also a notable figure in the field of chess endgame studies. He composed around 200 studies, and was co-inventor of the Guy-Blandford-Roycroft code for classifying studies. He also served as the endgame study editor for the British Chess Magazine from 1948 to 1951.
Guy wrote four papers with Paul Erdős, giving him an Erdős number of 1. He also solved one of Erdős problems.
His son, Michael Guy, is also a computer scientist and mathematician. *Wik

1918 Leslie Fox (30 September 1918 – 1 August 1992) was a British mathematician noted for his contribution to numerical analysis. *Wik


1953 Lewis Fry Richardson, FRS (11 October 1881 - 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist and pacifist who 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 on fractals and a method for solving a system of linear equations known as modified Richardson iteration.
*Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
~Lewis Richardson

1985 Dr. Charles Francis Richter (26 Apr 1900, 30 Sep 1985) was an American seismologist and inventor of the Richter Scale that measures earthquake intensity which he developed with his colleague, Beno Gutenberg, in the early 1930's. The scale assigns numerical ratings to the energy released by earthquakes. Richter used a seismograph (an instrument generally consisting of a constantly unwinding roll of paper, anchored to a fixed place, and a pendulum or magnet suspended with a marking device above the roll) to record actual earth motion during an earthquake. The scale takes into account the instrument's distance from the epicenter. Gutenberg suggested that the scale be logarithmic so, for example, a quake of magnitude 7 would be ten times stronger than a 6.*TIS

2014 Martin Lewis Perl (June 24, 1927 – September 30, 2014) was an American physicist who won the Nobel Prize in Physics in 1995 for his discovery of the tau lepton.
He received his Ph.D. from Columbia University in 1955, where his thesis advisor was I.I. Rabi. Perl's thesis described measurements of the nuclear quadrupole moment of sodium, using the atomic beam resonance method that Rabi had won the Nobel Prize in Phyics for in 1944.
Following his Ph.D., Perl spent 8 years at the University of Michigan, where he worked on the physics of strong interactions, using bubble chambers and spark chambers to study the scattering of pions and later neutrons on protons.[1] While at Michigan, Perl and Lawrence W. Jones served as co-advisors to Samuel C. C. Ting, who earned the Nobel Prize in Physics in 1976.
Seeking a simpler interaction mechanism to study, Perl started to consider electron and muon interactions. He had the opportunity to start planning experimental work in this area when he moved in 1963 to the Stanford Linear Accelerator Center (SLAC), then being built in California. He was particularly interested in understanding the muon: why it should interact almost exactly like the electron but be 206.8 times heavier, and why it should decay through the route that it does. Perl chose to look for answers to these questions in experiments on high-energy charged leptons. In addition, he considered the possibility of finding a third generation of lepton through electron-positron collisions. He died after a heart attack at Stanford University Hospital on September 30, 2014 at the age of 87. *Wik

2017 Vladimir Voevodsky (Jun 4, 1966 - Sep 30, 2017) formerly a gifted but restless student who flunked out of college out of boredom before emerging as one of the most brilliant and revolutionary mathematicians of his generation, died on Sept. 30 at his home in Princeton, N.J. He was 51.

Dr. Voevodsky was renowned for founding entirely new fields of mathematics and creating groundbreaking new tools for computers to confirm the accuracy of proofs. In 2002, he was awarded the Fields Medal, which recognizes brilliance and promise in mathematicians under 40.
He was “one of the giants of our time,” Thomas Hales, a mathematician at the University of Pittsburgh, said in an interview. Dr. Voevodsky, he said, transformed every field he touched. In his work using computers, for example, he upended mathematical thinking to such a degree that he changed the meaning of the equals sign.

He added: “His ideas gave a new way for all mathematicians to do what they do, a new foundation. The foundations of math are like a constitutional document that spells out the governing rules all mathematicians agree to play by. He has given us a new constitution.”

Vladimir Voevodsky was born in Moscow. His father, Alexander, directed a laboratory in experimental physics at the Russian Academy of Sciences; his mother, Tatyana Voevodskaya, was a chemistry professor at Moscow University. *obit NYTimes

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Friday, 29 September 2023

The Harmony of the Harmonic Mean


Things happen in threes according to the old myth, and in this case it was true. I was doing some research on the early history of a mathematical problem often called the "cistern" problem. You probably know the type; "If one pipe can fill a cistern in 6 hours and another can fill it in four hours, how long would it take both pipes working together." While I was working on that, I got a nice article sent to me on the first proof that the harmonic sequence diverges... and then, I was reading a blog by Dave Marain Math Notationsin which he posed a problem that asked, in its general form, given a square inscribed in a right triangle (with one corner at the right angle of the triangle), what is the length of a side of the square in terms of the legs of the triangle.

So what do all these have in common with each other. dare I say what makes them in "harmony"?.... the answer is Harmony, or at least the mathematical relationship of the harmonic mean.

To the early Greeks, if Nichomachus can be believed, all the means were descriptive of musical relations. Much is often made of the Harmonic Mean in relation to a musical sense, but this may not represent the Greek view. Euclid used the word enarmozein to describe a segment that just fits in a given circle. The word is a form of the word Harmozein which the more competent Greek Scholars tell me means to join or to fit together. Jeff Miller's Web site on the first use of Mathematical terms contains a reference to the very early origin of the harmonic mean, 'A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, 'There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic.' The term harmonic mean was also used by Aristotle. "
My search for the early roots of the cistern problem had taken me back to Heron's Metre'seis around the year fifty of the common era. The problem became a staple in arithmetics and problem books and was used by Alcuin (775) and appears in the Lilavati of Bhaskara (1150). I found the illustration I used on the blog for The First Illustrated Arithmetic a few days ago, from the 1492 arithmetic, Trattato di aritmetica by Filippo Calandri.

The solution to a cistern problem is the harmonic mean of the times taken by each pipe. For example, one problem asks "If one pipe can fill a cistern in three hours, and a second can fill it in five hours, how fast will the two pipes take to fill the cistern if both are opened at once. The solution is given by finding the average rate of fill of the two rates, the harmonic mean of three and five, which is three and three-quarter hours. But as the name "mean" suggest, that's the average rate of the two so working together, they would take one-half the time, one and seven-eighths hours, or about an hour and 53 minutes.

The Harmonic mean is the reciprocal of the mean of the reciprocals of the values, so for values a and b, the harmonic mean is given bywhich for two numbers can be simplified to the more economical
Heron might have been the first recorded example of a cistern problem, but a problem calling on the reader to use the harmonic mean occurs even earlier in the Rhind Mathematical Papyrus, now located in the British Museum, in problem 76. The problem involves making loaves of bread with different qualities, but the solution is still the harmonic mean. (I have learned from David Singmaster's Chronology of Recreational Mathematics that the cistern problem appeared, perhaps 300 years before Heron's use, in China by Chiu Chang Suan Shu (around 150 BC).

The series of terms formed by the reciprocals of the positive integers is a common torment for college students in their first introduction to analysis. The sequencein which each number gets smaller and smaller seems to very slowly approach some upper limit. Even after adding 250,000,000 terms, the sum is still less than twenty, and yet... in the mid 1300's, Nichole d'Oresme showed that it will eventually pass any value you can name. In short, it diverges, slowly, very, very slowly, to infinity. Even when warned, it seems like students want to believe it converges. A well-known anecdote about a teacher trying to get student's to remember that it diverges goes:

"Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”<\blockquote>

By the way, each number in the harmonic series is the harmonic mean of the numbers on each side of it (so 1/2 and 1/4 have a harmonic mean of 1/3), and in fact, of any numbers equally spaced away from it such as 1 and 1/5 also have a harmonic mean of 1/3.

And then, I came across that little problem of a square inscribed in a right triangle. If the two legs are a and b, then the sides of the square will have a length equal to the one-half the harmonic mean of a and b .  More generally, a square inscribe in any triangle with one side along a base will have sides equal to one half the harmonic mean of the base and the altitude to that base.  There are lots of other interesting problems that yield to the use of the harmonic mean, and I mean to write again on that collection.

So I guess things do come in threes, unless I come across another one, but whether it comes in threes or fours, it all seems to work together, in perfect harmony.

Many students who struggle with a different puzzle type problem might want to investigate how it too, relates to the harmonic mean, the one where they ask, If you drive to grandmother's house at 60 miles per hour and drive home at 40 miles per hour, what was your average speed for the round trip?  There are dozens more, so just to get a collection, send your favorite problem related to the harmonic mean, and I'll update as they come along.