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 Tet_{n} = \( \frac {(n)(n+1)(n+2)}{6}\)

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.

See more Extended math facts for every year date here.

**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^{-1}*x* = *x* - *x*^{3}/3 + *x*^{5}/5 - *x*^{7}/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^{-1}*x* = *x* - *x*^{3}/3 + *x*^{5}/5 - *x*^{7}/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.

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**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,

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

**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

**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 Karl Petr** He worked in analytic number theory, algebraic equations, and invariant theory. *VFR

**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.*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 generalisations of the sieve methods of Viggo Brun. In 1986 he won the Wolf Prize. *TIS

**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)

**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

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

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