Wednesday 4 September 2024

On This Day in Math - September 4

The science of pure mathematics may claim to be the most original creation of the human spirit.

The 247th day of the year; 247 is the smallest number which can be expressed as the difference between two integers such that together, they contain all digits 0-9. (spoilerthe two numbers are written at the bottom of this post)

The digits of 247 sum to its smallest prime factor (247 = 13 x 19 and 2 + 4 + 7 = 13) *Prime Curios (How many of the composite days of the year sum to one of their prime factors?)

247 is the 13th Pentagonal number ( n x (3n-1)/2) . The average of the first n pentagonal numbers, is the nth Triangular number. Like all pentagonal numbers, it is the sum of the n consecutive numbers starting with n. so the 13th pentagonal number is 13 + 14 + 15 + ... 25 = 247

EVENTS

1675 John Collins, after mentioning Tschirnhaus in a letter to James Gregory, writes: “... there being present with him a Dane named George Moorh who lately published in low Dutch, two little Books the one named Euclides Danicus where he pretends to perform all Euclids problems with a paire of Compasses only without Ruler, and another entitled Euclides Curiosus, wherein with a Ruler and a forke (or the Compasses at one opening) he performs the same ... ” See MT 53(1960), 127–132. (Mohr was a friend of Tschirnhaus, and he spent his last few years as a guest in his house.)

This book, proving the Mohr–Mascheroni theorem 125 years earlier than Lorenzo Mascheroni, would languish in obscurity until its rediscovery in 1928.

 *Wik

1740, Philip Naudé the younger (1684-1747) wrote Euler from Berlin to ask “how many ways can the number 50 be written as a sum of seven different positive integers?” The problem seems to have captured Euler’s imagination. Euler gave his first answer on April 6, 1741, in a paper he read at the weekly meeting of the St. Petersburg Academy. (Wm Dunham says that Euler replied by mail within a few days, and apologized for the delay as he had been suffering from poor eyesight.) That paper was published ten years later and is number 158 on Eneström’s index. Euler solved the problem in a different way in the Introductio in analysin infinitorum, E101,published in 1748, and made more improvements in a paper De partitione numerorum, “On the partition of numbers,” E191, written in 1750, published in 1753. Late in his life, in 1769, he returned to the problem in De partione numerorum in partes tam numero quam specie datas, “On the partition of 2 numbers into a given kind or number of parts,” E394, published in 1770. *Ed Sandifer, How Euler Did It ( A single letter revisited and the solution expanded over a thirty year period. Euler was a little of a mathematical bulldog.)(Euler got 522 for the answer to Naude's question)

Restricted partitions satisfy some additional condition. Notable among these are distinct partitions, where each summand is different, and odd partitions, where each summand is odd. For each positive number, the number of partitions with odd parts is equal to the number of partitions with distinct parts, denoted by {d(n)}. This result was proved by Leonhard Euler in 1748.

1749  Voltaire wrote to one of Émilie du Châtelet's friends :-

Mme du Châtelet informs you that this night, being at her desk working on Newton, she felt a little call. The little call was a daughter, who appeared in an instant. She was laid on a quarto book of geometry. The mother has gone to lie down and, if she were not asleep, she would be writing to you.

Sadly, du Châtelet died 6 days later.

*MacTutor, SAU

Portrait by Maurice Quentin de La Tour

1751 Leonhard Euler writes to Christian Goldbach with a conjecture about the number of dissections of a polygon into triangles by using diagonals.  He would continue to work on the problem and would eventually share the concept with Johann Andreas von Segner. In the Goldbach letter, Euler gave a product formula for the seqeunce that is now commonly called the Catalan Numbers.$\frac{2 * 6 * 10 \dots (4n-10}{2*3*4 \dots (n-1}$. Segner was the first to give the recursive formula .

The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.

The name “Catalan numbers” originated from John Riordan  (April 22, 1903 – August 27, 1988).

In 1988, it came to light that the Catalan number sequence had been used in China by the Mongolian mathematician Mingantu by 1730

1821 On  September 4th, 1821, Michael Faraday discovered that a vertically mounted wire carrying an electric current would rotate continuously round a magnet sticking out of a bowl of mercury. He named this phenomenon electro-magnetic rotations. *engineering-timelines.com

1893 “The proof of the transcendency of π will hardly diminish the number of circle-squarers, however; for this class of people has always shown an absolute distrust of mathematicians and a contempt for mathematics that cannot be overcome by any amount of demonstration.” Felix Klein in The Evanston Colloquium. Lectures on Mathematics (1894), pp. 52–53. *VFR

In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim refuted by John Wallis as part of the Hobbes–Wallis controversy. During the 18th and 19th century, the false notions that the problem of squaring the circle was somehow related to the longitude problem, and that a large reward would be given for a solution, became prevalent among would-be circle squarers.

Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined 𝜋 as equal to 3.2. Goodwin then proposed the Indiana pi bill in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.

 Felix Klein

1899 A 1904 academic calendar marked this day as the day Dedekind died. He wrote the publisher saying that while 4 September might be correct, 1899 certainly was not, for on that day he had enjoyed a stimulating mathematical discussion with his dinner guest and honored friend, Georg Cantor. *VFR

 Dedekind

1963 India issued a stamp honoring Dadabhoy Naoroji (1825–1917), mathematician and stateman. [Scott #376]. *VFR

1988 Only a few seconds before ignition, a computer halts an engine test in preparation for the launch of the space shuttle Discovery. The shuttle engine's computerized controllers determined that a valve was not closing fast enough and sent a major component failure command from the computer to all three engines, telling them not to fire. The test and computer system were part of NASA efforts to ensure the safety of Discovery, whose flight would be the first since the Challenger explosion in 1986. *CHM

BIRTHS

973 Al-Biruni born (born 5 September 973 in Kath, Khwarezm, now region in Uzbekistan, died 13 December 1048 in Ghazni). He wrote 15 works on mathematics, three of which are extant.*VFR
Al-Biruni is regarded as one of the greatest scholars of the medieval Islamic era and was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist and linguist. He was conversant in Chorasmian, Persian, Arabic, Sanskrit and Turkic, and also knew Greek, Hebrew and Syriac. He spent a large part of his life in Ghazni in modern-day Afghanistan, capital of the Ghaznavid dynasty which ruled eastern Iranian lands and the northwestern Indian subcontinent. In 1017 he traveled to the Indian subcontinent and became the most important interpreter of Indian science to the Islamic world. He is given the titles the "founder of Indology" and the "first anthropologist". He was an impartial writer on custom and creeds of various nations, and was given the title al-Ustdadh ("The Master") for his remarkable description of early 11th-century India. He also made contributions to Earth sciences, and is regarded as the "father of geodesy" for his important contributions to that field, along with his significant contributions to geography.

In his Codex Masudicus (1037), Al-Biruni theorized the existence of a landmass along the vast ocean between Asia and Europe, or what is today known as the Americas. He argued for its existence on the basis of his accurate estimations of the Earth's circumference and Afro-Eurasia's size, which he found spanned only two-fifths of the Earth's circumference, reasoning that the geological processes that gave rise to Eurasia must surely have given rise to lands in the vast ocean between Asia and Europe. He also theorized that at least some of the unknown landmass would lie within the known latitudes which humans could inhabit, and therefore would be inhabited.*Wik

The statue of Al-Biruni in United Nations Office in Vienna

1809 Luigi Menabrea (September 4, 1809 – May 24, 1896) was a French-born soldier and engineer who made contributions to elasticity theory and became prime-minister of Italy.*SAU

1848 Heinrich Bruns (4 Sept 1848 in Berlin, Germany - 23 Sept 1919 in Leipzig, Germany) was interested in astronomy, mathematics and geodesy and worked on the three body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which the coefficients of the time depend..*SAU

1899 Hildegard Rothe-Ille, Hildegard Ille, (Sep 4, 1899– Dec 1, 1942), was a German mathematician.

She was one of Issai Schur’s doctoral students. According to Alexander Soifer, “Van der Waerden walked away from Ramseyan prehistory. Issai Schur, on the other hand, continued to produce Ramseyan mathematics, and moreover directed and inspired his PhD students Richard Rado, Hildegard Ille and Alfred Brauer to do the same.”

She received her doctorate in mathematics in 1924. Beginning on April 1, 1925, she was a year-long scholarship holder at the Kaiser Wilhelm Institute for Physics, which was headed by Albert Einstein at the time. She was the only woman to receive a scholarship from that institute in that academic year, and she received a higher scholarship than her male counterparts did.

She taught at the Chamisso school in Berlin-Schöneberg from 1926 until 1928. After marrying in 1928, due to German law she was not allowed to work for pay; however, she did review papers about mathematics. Under the name Hildegard Rothe she reviewed 40 papers which had been published between 1926 and 1928, and under the name Hildegard Rothe-Ille she reviewed 129 papers which had been published between 1930 and 1937, before having to flee from the Nazi regime in 1937.

The 1940 United States census records that she was a part-time teacher of German at William Penn College.*Wik

1906 Max Ludwig Henning Delbrück (September 4, 1906 – March 9, 1981)
Delbrück was a German-American biophysicist and Nobel laureate.
Delbrück studied astrophysics, shifting towards theoretical physics, at the University of Göttingen. After receiving his Ph.D. in 1930, he traveled through England, Denmark, and Switzerland. He met Wolfgang Pauli and Niels Bohr, who got him interested in biology.
In 1937, he moved to the United States to pursue his interests in biology, taking up research in the Biology Division at Caltech on genetics of the fruit fly Drosophila melanogaster.
Delbrück was one of the most influential people in the movement of physical scientists into biology during the 20th century. Delbrück's thinking about the physical basis of life stimulated Erwin Schrödinger to write the highly influential book, What Is Life?. Schrödinger's book was an important influence on Francis Crick, James D. Watson and Maurice Wilkins who won a Nobel prize for the discovery of the DNA double helix. *TIA

1927 John McCarthy (September 4, 1927 – October 24, 2011) was an American computer scientist and cognitive scientist. He was one of the founders of the discipline of artificial intelligence. He co-authored the document that coined the term "artificial intelligence" (AI), developed the programming language family Lisp, significantly influenced the design of the language ALGOL, popularized time-sharing, and invented garbage collection.

McCarthy spent most of his career at Stanford University. He received many accolades and honors, such as the 1971 Turing Award for his contributions to the topic of AI, the United States National Medal of Science, and the Kyoto Prize.

DEATHS

1784 César-François Cassini de Thury Died (17 Jun 1714, 4 Sep 1784)French astronomer and geodesist (Cassini III), who continued surveying work he began while assisting his father, Jacques Cassini (Cassini II), resulting in the first topographical map of France produced by modern principles. His grandfather, Giovanni Domenico Cassini (Cassini I) discovered four satellites of Saturn, a band on planet's surface, and that its ring was subdivided. Cassini I was the first to assume effective direction (1671) of the new observatory established by the Académie Royale des Sciences in Paris, which his descendants in turn continued. Cassini III was the first official director of the observatory when the post was created by the king in 1771. His son was Jean-Dominique Cassini (Cassini IV).*TIS He produced the first reliable maps of France. *SAU

1822 Josiah Meigs (August 21, 1757 – September 4, 1822) was an American academic, journalist and government official meteorologist and mathematician, born.*Wik This freethinking Democrat left his professorship at Yale for political reasons and became president of the University of Georgia. He applied Galileo’s formula for fallen bodies to the nine day’s fall of Lucifer and his angels, to determine that Hell was 1,832,308,363 miles deep. [Struik, Origins of American Science, p. 370] *VFR

1881 George Palmer Williams (Woodstock, Vermont, April 13, 1802-Ann Arbor, September 4, 1881) He graduated Bachelor of Arts from the University of Vermont in 1825, and then studied about two years in the Theological Seminary at Andover, Massachusetts. He did not complete the course, but took up teaching, which proved to be his life work.
He was Principal of the Preparatory School at Kenyon College, Ohio, from 1827 to 1831. In 1831 he was elected to the chair of Ancient Languages in the Western University of Pennsylvania, but after two years he returned to Kenyon College, where he remained until called, in 1837, to the branch of the incipient University of Michigan at Pontiac.
In 1841, when the College proper was opened at Ann Arbor, he was made Professor of Natural Philosophy. In 1854 he was transferred to the chair of Mathematics and in 1863 to the chair of Physics. From 1875 to 1881 he was Emeritus Professor of Physics.
He received the degree of Doctor of Laws from Kenyon College in 1849. The University Senate in a memorandum relative to his death declared that: "Dr. Williams welcomed the first student that came to Ann Arbor for instruction; as President of the Faculty he gave diplomas to the first class that graduated, and from the day of his appointment to the hour of his death his official connection with the University was never broken."
In 1846 he was ordained to the ministry of the Protestant Episcopal Church; but he did no regular parish work, except for a short time in Ann Arbor. He was first and last a teacher, beloved by his colleagues and pupils and universally respected and honored.
Some years before his death the alumni raised a considerable fund, the proceeds of which were to be paid to him during his lifetime and after his death were to be used for maintaining a professorship named in honor of his memory. *Hinsdale and Demmon, History of the University of Michigan 221

1969 Marcel Riesz died (16 November 1886 – 4 September 1969) His interests ranged from functional analysis to partial differential equations, mathematical physics, number theory and algebra. Later in his career he also worked on Clifford algebras and spinors. The first period of his work, from the beginning of his doctoral research up to around the beginning of World War I, concentrated on the theory of series, in particular the summability theory of power series, trigonometric series and Dirichlet series. In 1914 he gave an interpolation formula for trigonometric polynomials. This was an important discovery and the formula now appears in most texts on interpolation. It leads to quick proofs of Bernstein's inequality and Markov's inequality. Another highlight from this period is his beautiful proof of Fatou's theorem which give conditions under which the power series of an analytic function converges to a point on its circle of convergence. *SAU

1984 Ernst Carl Gerlach Stueckelberg (February 1, 1905, September 4, 1984) was a Swiss mathematician and physicist. In 1938 he recognized that massive electrodynamics contains a hidden scalar, and formulated an affine version of what would become known as the Abelian Higgs mechanism. He also proposed the law of conservation of baryon number. In 1953 he and the mathematician Andre Petermann discovered the renormalization group.
He was awarded the Max Planck medal.*Wik

1996  Joan Elisabeth Lowther Murray, MBE (née Clarke; 24 June 1917 – 4 September 1996) was an English cryptanalyst and numismatist who worked as a code-breaker at Bletchley Park during the Second World War. Although she did not personally seek the spotlight, her role in the Enigma project that decrypted the German secret communications earned her awards and citations, such as appointment as a Member of the Order of the British Empire (MBE), in 1946.

Clarke and Turing had been close friends since soon after they met, and continued to be until Turing's death in 1954. They shared many hobbies and had similar personalities. They became very good friends at Bletchley Park. Turing arranged their shifts so they could work together, and they also spent much of their free time together. In early 1941, Turing proposed marriage to Clarke, and subsequently introduced her to his family. Although he privately admitted his homosexuality to her—she was reportedly unfazed by the revelation—Turing decided that he could not go through with the marriage, and broke up with Clarke in mid-1941. Clarke later admitted that she suspected Turing's homosexuality for some time, and it was not much of a surprise when he made the admission to her.

2011 Hans Grauert (8 February 1930 in Haren, Emsland, Germany – 4 September 2011) was a German mathematician. He is known for major works on several complex variables, complex manifolds and the application of sheaf theory in this area, which influenced later work in algebraic geometry. Together with Reinhold Remmert he established and developed the theory of complex-analytic spaces. He became professor at the University of Göttingen in 1958, as successor to C. L. Siegel. The lineage of this chair traces back through an eminent line of mathematicians: Weyl, Hilbert, Riemann, and ultimately to Gauss. Until his death, he was professor emeritus at Göttingen.   Grauert was awarded a fellowship of the Leopoldina. *Wik

247 = 50123 - 49876

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