## Friday 14 June 2024

### On This Day in Math - June 14

It would be very discouraging if
somewhere down the line
you could ask a computer if
the Riemann hypothesis is correct
and it said, 'Yes, it is true,
but you won't be able to understand the proof.

~Ronald Graham

The 165th day of the year; 165 is a tetrahedral number, and the sum of the first nine triangular numbers. The tetrahedral numbers are found on the fourth diagonal of Pascal's Arithmetic Triangle, and given by the combinations of (n+2 Choose 3) or Tetn = $\frac {(n)(n+1)(n+2)}{6}$

165 is sort of a prime average(or an average of primes) The two nearest primes are 163 and 167, with 165 as their average; The next two nearest are 157 and 173, yeah, 165 is their average; The next two out are 179 and 151, yes again, average is 165; then 149 and 181, yep!.... 139 and 191, yep!.... 137 and 193...Oh Yeah!...

165 is a sphenic number, the product of three distinct primes.

165 is also the sum of the squares of the first five odd numbers. 165 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 , it's factors, 165 =3 x 5 x 11, but is more interesting as (9 x 10 x 11) / 6.  Try the sum of the first seven odd squares.  Can you apply this to sum of first six or eight?

165 is the sum of the divisors of the first fourteen integers.

EVENTS

1444 Nilakantha was a mathematician and astronomer from South India who wrote texts on both astronomy and infinite series. The series π/4 = 1 - 1/3 + 1/5 - 1/7 + ... is a special case of the series representation for arctan, namely
tan-1x = x - x3/3 + x5/5 - x7/7 + ...
It is well known that one simply puts x = 1 to obtain the series for π/4. Ranjan  Roy from Beloit College reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s. The contributions of the two European mathematicians to this series are well known but in his paper, Roy mentions the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed.
Nilakantha derived the series expansion tan-1x = x - x3/3 + x5/5 - x7/7 + ... by obtaining an approximate expression for an arc of the circumference of a circle and then considering the limit. An interesting feature of his work was his introduction of several additional series for π/4 that converged more rapidly than π/4 = 1 - 1/3 + 1/5 - 1/7+ ... .;*SAU

His best known work, written on palm leaves, is called Tantrasangraha [or Tantrasamgraha] (1501).

 *MAA

1564 The mathematician/magician John Dee returns to England after five years on the continent and presents his new book, Monas Hieroglyphica, to Queen Elizabeth. The book provides Dees conjecture that the astronomical planet symbols were relics of a lost universal language. He also stated that all the symbols could be combined together into a single symbol, or monad, which was a variant on the sign for Mercury. The monad appears in the center of the book's frontispiece He felt this union exemplified the unity of the universe. *Benjamin Wolley, The Queen's Conjuror

 *Wik

1648 Margaret Jones is hanged in Boston for witchcraft in the first such execution for the Massachusetts colony. * The Great Geek Manual  (Witches were not the first to be hanged in Massachusetts.  John Billington, a colonist who arrived on the Mayflower, was the first person executed in Massachusetts, in 1630. He was hung for killing John Newcomen. Others were hung for religious differences. William Robinson and Marmaduke Stevenson, two Quakers who came from England in 1656 to escape religious persecution, are executed in the Massachusetts Bay Colony for their religious beliefs.

1649 John Wallis appointed as Savilian professor of geometry at Oxford.
Wallis created the term "continued fraction" and popularized the ∞ symbol for infinity. One aspect of Wallis's mathematical skills has not yet been mentioned, namely his great ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. One night he calculated in his head the square root of a number with 53 digits. In the morning he dictated the 27-digit square root of the number, still entirely from memory. It was a feat that was rightly considered remarkable, and Henry Oldenburg, the Secretary of the Royal Society, sent a colleague to investigate how Wallis did it. It was considered important enough to merit discussion in the Philosophical Transactions of the Royal Society of 1685.

1770   D/1770 L1, popularly known as Lexell's Comet after its orbit computer Anders Johan Lexell, was a comet discovered by astronomer Charles Messier in June 1770. It is notable for having passed closer to Earth than any other comet in recorded history, approaching to a distance of only 0.015 astronomical units (2,200,000 km; 1,400,000 mi), or six times the distance from the Earth to the Moon. The comet has not been seen since 1770 and is considered a lost comet.

 Charles Messier, who discovered Lexell's Comet

1777 the Continental Congress approved the design of a national flag. Since 1916, when President Woodrow Wilson issued a presidential proclamation establishing a national Flag Day on June 14, Americans have commemorated the adoption of the Stars and Stripes by celebrating June 14 as Flag Day. Prior to 1916, many localities and a few states had been celebrating the day for years. Congressional legislation designating that date as the national Flag Day was signed into law by President Harry Truman in 1949; the legislation also called upon the president to issue a flag day proclamation every year.
According to legend, in 1776, George Washington commissioned Philadelphia seamstress Betsy Ross to create a flag for the new nation. Scholars debate this legend, but agree that Mrs. Ross most likely knew Washington and sewed flags. To date, there have been twenty-seven official versions of the flag, but the arrangement of the stars varied according to the flag-makers' preferences until 1912 when President Taft standardized the then-new flag's forty-eight stars into six rows of eight. The forty-nine-star flag (1959-60), as well as the fifty-star flag, also have standardized star patterns. The current version of the flag dates to July 4, 1960, after Hawaii became the fiftieth state on August 21, 1959. *Library of Congress, On This Day in History (The 48 and 50 star flags are related to an  interesting packing problem. See  The Very Mathematical US Flag Starfield

1822 Charles Babbage read a paper to the Astronomical Society of London entitled “Note on the application of machinery to the computation of astronomical and mathematical tables.” He announced the successful completion of a “Diﬀerence engine,” the forerunner of our modern computers. See Dubbey, The Mathematical Work of Charles Babbage, p. 175. *VFR

Babbage began to construct a small difference engine in 1819 and had completed it by 1822. He announced his invention in a paper Note on the application of machinery to the computation of astronomical and mathematical tables read to the Royal Astronomical Society on 14 June 1822.

Although Babbage envisaged a machine capable of printing out the results it obtained, this was not done by the time the paper was written. An assistant had to write down the results obtained. Babbage illustrated what his small engine was capable of doing by calculating successive terms of the sequence n^2 + n + 41.

The terms of this sequence are 41, 43, 47, 53, 61, ... while the differences of the terms are 2, 4, 6, 8, .. and the second differences are 2, 2, 2, ..... The difference engine is given the initial data 2, 0, 41; it constructs the next row 2, (0 + 2), [41 + (0 + 2)], that is 2, 2, 43; then the row 2, (2 + 2), [43 + (2 + 2)], that is 2, 4, 47; then 2, 6, 53; then 2, 8, 61; ... Babbage reports that his small difference engine was capable of producing the members of the sequence at the rate of about 60 every 5 minutes. *SAU

A difference engine was constructed in the 1980's at the Science Museum Library in London. Once completed, both the engine and its printer worked flawlessly, and still do. The difference engine and printer were constructed to tolerances achievable with 19th-century technology,

In 1834, the first U.S. patent for a practical underwater diving suit was issued to Leonard Norcross of Dixfield, Maine (No. X8255).* Calling it a “Diving Armor,” he designed an airtight outfit made from India rubber and leather. It had a brass helmet connected via a rubber hose to an air bellows pump on a boat. To reduce buoyancy, the feet were weighted with lead shot. In May 1834, one month earlier, he tested the diving suit in the Webb River. Norcross named his son Submarinus in honor of the achievement.* The first truly effective diving suit with pump is attributed to an Englishman, Augustus Siebe, who designed it in 1829 and was entrusted with equipping the French Navy until 1857.*

1937 Nobel Winning research rejected by Nature Magazine. The editor of Nature sent the rejection letter below to Hans Kreb. The paper rejected explained his recent discovery of the citric acid cycle, or “Krebs cycle”. He next submitted the paper to the journal Enzymologia in Holland, where it was accepted. Krebs would win the 1953 Nobel Prize for that research. *The Scientist (w/ HT to Ben Gross)

1951 UNIVAC I, the ﬁrst commercial electronic computer, was demonstrated and dedicated at the Bureau of the Census at Philadelphia. It could accept information from magnetic tape at the rate of 10,000 characters per second, yet could retain a maximum of 1000 numbers. *VFR This "first" statement is often repeated, but I now know of at least two earlier claimants for the title. Wikipedia has (in two different places):
The Ferranti Mark 1, also known as the Manchester Electronic Computer in its sales literature,and thus sometimes called the Manchester Ferranti, was the world's first commercially available general-purpose electronic computer. It was "the tidied up and commercialized version of the Manchester computer". *Wik
In addition, there was The first commercial computer in the world was the BINAC built by the Eckert–Mauchly Computer Corporation and delivered to Northrop Aircraft Company in 1949.*Wik

1956 Fred Reines and Clyde Cowan send a telegram to Wolfgang Pauli from Los Alamos, "We are happy to tell you that we have definitely detected neutrinos from fission fragments by observing inverse beta decay of protons." Pauli's famous reply, "Everything comes to him who knows how to wait." *Charles P. Enz, No Time to be Brief: A Scientific Biography of Wolfgang Pauli

BIRTHS

1736 Charles-Augustin de Coulomb (14 June 1736 – 23 August 1806) was a French physicist best known for the formulation of Coulomb's law, which states that the force between two electrical charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Coulombic force is one of the principal forces involved in atomic reactions. The inverse-square relationship is also seen in the relationship of the gravitation force between masses. In 1777, he invented a torsion balance which he subsequently modified for electrical measurements. He also did research on friction of machinery, on windmills, and on the elasticity of metal and silk fibres.*TIS

1832 Nikolaus August Otto (14 June 1832, Holzhausen an der Haide, Nassau - 26 January 1891, Cologne)born. German engineer who developed the four-stroke internal-combustion engine, which offered the first practical alternative to the steam engine as a power source. A French engineer, Alphonse Beau de Rochas, formulated the basic design for the four-stroke internal combustion engine and patented it in 1862, but never built a working model. In 1876, Otto used principles from Beau de Rochas and others to construct the prototype of today's automobile engines, often called the Otto-cycle engine. He sold thousands of copies before Beau de Rochas sued him and invalidated Otto's patent. But light, efficient Otto-cycle engines largely enabled the creation of automobiles, powerboats, motorcycles and even airplanes. *TIS

Father of the ottomobile.........(sorry😓😒

1856 Andrey Andreyevich Markov(14 June 1856 N.S. – 20 July 1922) Russian mathematician who helped to develop the theory of stochastic processes, especially those called Markov chains, sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work based on the study of the probability of mutually dependent events has been developed and widely applied to the biological and social sciences. *TIS

1868 Karel Petr (14 June 1868, Zbyslav, Austria-Hungary – 14 February 1950, Prague, Czechoslovakia) was a Czech mathematician. He was one of the most renowned Czech mathematicians of the first half of the 20th century.

Petr is known for the Petr–Douglas–Neumann theorem in plane geometry, which he proved in 1908, and was independently rediscovered by Jesse Douglas in 1940 and Bernhard Neumann in 1941.

The Petr–Douglas–Neumann theorem asserts the following.   If isosceles triangles with apex angles 2kπ/n, for an integer k with 1 ≤ k ≤ n − 2 are erected on the sides of an arbitrary n-gon A0, whose apices are the vertices of a new n-gon A1, and if this process is repeated n-2 times, but with a different value of k for the n-gon formed from the free apices of these triangles at each step, until all values 1 ≤ k ≤ n − 2 have been used (in arbitrary order), to form a sequence A1, A2, ... An-2, of n-gons, their centroids all coincide with the centroid of A0, and the last one, An−2 is a regular n-gon .

For a triangle, this means n=3 and n-2 = 1, so only one set of isosceles triangles is needed to form the equilateral triangle. This is the well known Napolean's Theorem.  Several papers have been written concerning this issue[8][9] which cast doubt upon the idea that Napoleon created it. The problem appears in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820.

Question 10. Three equilateral triangles are thus constructed on the sides of a given triangle, A, B, D, the lines joining their centres, C, C', C" form an equilateral triangle. [The accompanying diagram shows the equilateral triangles placed outwardly.]

Question 11. If the three equilateral triangles are constructed as in the last figure, the lines joining their centres will also form an equilateral triangle. [The accompanying diagram shows the equilateral triangles places inwardly.]

Question 12. To investigate the relation between the area of the given triangle and the areas of these two equilateral triangles.

Petr–Douglas–Neumann theorem as applied to a quadrilateral A0 = ABCD. A1 = EFGH is constructed using apex angle π/2 and A2 = PQRS with apex angle π.

*Wik

1903 Alonzo Church(June 14, 1903 – August 11, 1995) made important contributions to mathematical logic and theoretical computer science. *SAU
The lambda calculus emerged in his famous 1936 paper showing the unsolvability of the Entscheidungsproblem. This result preceded Alan Turing's famous work on the halting problem, which also demonstrated the existence of a problem unsolvable by mechanical means. Church and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative "mechanical processes for computation." This resulted in the Church–Turing thesis.
The lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor. *Wik

1910 Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.

John published his first paper in 1934 on Morse theory. He was awarded his doctorate in 1934 with a thesis entitled Determining a function from its integrals over certain manifolds from Göttingen. With Richard Courant's assistance he spent a year at St John's College, Cambridge. During this time he published papers on the Radon transform, a theme to which he would return throughout his career.*Wik

1917 Alte Selberg (14 June 1917, Langesund, Norway – 6 August 2007) Norwegian-born mathematician who is one of the foremost analytic number theorists. After working in isolation during WW II, due to the occupation of Norway by the Nazis, his accomplishments in the theory of the Riemann zeta function became known. During the 1950's he developed the Selberg trace formula, his most famous accomplishment. It establishes a duality between the length spectrum of a Riemann surface and the eigenvalues of the Laplacian which is analogous to the duality between the prime numbers and the zeros of the zeta function. He was awarded the Fields Medal in 1950 for his work in number theory on generalizations of the sieve methods of Viggo Brun. In 1986 he won the Wolf Prize. *TIS

1935 Louise Schmir Hay (June 14, 1935 – October 28, 1989) was a French-born American mathematician. Her work focused on recursively enumerable sets and computational complexity theory, which was influential with both Soviet and US mathematicians in the 1970s. When she was appointed head of the mathematics department at the University of Illinois at Chicago, she was the only woman to head a math department at a major research university in her era.

Her work was influential with both Soviet and US mathematicians of the period. She co-founded the Association for Women in Mathematics (AWM) in an effort to provide support to other working mothers. In 1978, she won a Fulbright Scholarship, as did her husband, and they spent the year studying in the Philippines. In 1979, Hay was named the acting head of the University of Illinois' mathematics department. n 1980, she was appointed to the executive board of the AWM and remained in that post until 1987. She was also named as secretary of the Association for Symbolic Logic in 1982.

DEATHS

1746 Colin MacLaurin(February 1698 – 14 June 1746) organized the defence of Edinburgh, Scotland, during the Jacobite rebellion. Due to the exertion and exposure he ruined his health and died on this date of edema. For the previous twenty years his main work was on ﬂuxions, although he was a popular lecturer on many subjects at the University of Edinburgh. *VFR His major work on the fluxions was in response to the attack on the calculus by Bishop Berkeley.

1768 James Short, (10 June O.S. (21 June N.S.) 1710 – 14 June 1768) British optician and astronomer who produced the first truly parabolic and elliptic (hence nearly distortionless) mirrors for reflecting telescopes. During his working life of over 35 years, Short made about 1,360 instruments - not only for customers in Britain but also for export: one is still preserved in Leningrad, another at Uppsala and several in America. Short was principal British collator and computer of the Transit of Venus observations made throughout the world on 6th June 1761. His instruments traveled on Endeavour with Captain Cook to observe the next Transit of Venus on 3rd June 1769, but Short died before this event took place.*TIS Died within one week of his birth date (10 June)

1875 Heinrich Louis d'Arrest (13 August 1822 – 14 June 1875) was a German astronomer, born in Berlin. His name is sometimes given as Heinrich Ludwig d'Arrest.
While still a student at the University of Berlin, d'Arrest was party to Johann Gottfried Galle's search for Neptune. On 23 September 1846, he suggested that a recently drawn chart of the sky, in the region of Urbain Le Verrier's predicted location, could be compared with the current sky to seek the displacement characteristic of a planet, as opposed to a stationary star. Neptune was discovered that very night.

D'Arrest's later work at the Leipzig Observatory led him, in 1851, to the discovery of the comet named for him (formally designated 6P/d'Arrest). He also studied asteroids, discovering 76 Freia, nebulae, and galaxies, discovering NGC 1 in 1861 and NGC 26 and NGC 358 in 1865.

In 1864 D'Arrest made an unsuccessful search for Martian satellites, and posited an upper limit of 70 minutes of arc as the distance from Mars within which a moon should be sought.

He won the Gold Medal of the Royal Astronomical Society in 1875.

In 1857, he married Auguste Emilie Möbius, daughter of his then-supervisor, August Ferdinand Möbius. He died in Copenhagen, Denmark.

1938 William Wallace Campbell (11 Apr 1862 near Findlay in Hancock county, Ohio; 14 Jun 1938 at age 76) American astronomer known particularly for his spectrographic determinations of the radial velocities of stars--i.e., their motions toward the Earth or away from it. In addition, he discovered many spectroscopic binary stars, and in 1924 he published a catalog listing more than 1,000 of them.*TIS
After a few years of local schooling he entered in 1882 the University of Michigan to study civil engineering, graduating Bachelor of Science in 1886. Whilst at university he developed his interest in astronomy when he read Simon Newcomb's Popular Astronomy.
After graduating he was appointed Professor of Mathematics at the University of Colorado but soon moved back to Michigan as an instructor in astronomy. In 1891 he was invited to work on spectroscopy at Lick Observatory in California. Campbell was a pioneer of astronomical spectroscopy and catalogued the radial velocities of stars. He was also recognized for his work in solar eclipse photography. In 1893 he discovered the Wolf–Rayet star HD 184738 (also known as Campbell's hydrogen envelope star). He was made a director of Lick Observatory from 1901 to 1930.

1946 Federigo Enriques (5 January 1871 – 14 June 1946) died in Rome. He was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry. *SAU

2008 Yurii Alekseevich Mitropolskiy (3 January 1917 — 14 June 2008) was a renowned Soviet, Ukrainian mathematician known for his contributions to the fields of dynamical systems and nonlinear oscillations.

He received his Ph.D. from Kyiv University, under the supervision of theoretical physicist and mathematician Nikolay Bogolyubov. Mitropolskiy is one of the most frequently joint-published mathematicians known, with at least 240 collaborators. Member of the Communist Party since 1945.
*Wik

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