**As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra.**

~Augustin-Louis Cauchy

There are 234 ways of grouping six children into rings of at least two children with one child at the center of each ring.

234 is the 55th abundant number. It's proper divisors are {1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117} and their sum is greater than 234, it is 312. There are 86 abundant nu\mbers in a non-leap year. 234=6(39) and all multiples two or greater of a perfect number are abundant.

234 begins a string of eight consecutive digits which is a prime number, 23456789. All the children need to know why there can not be any ordering of all nine digits , 1 to 9 that is prime. Teach one today.

Casting out nines is as old as Iamblichus, and as new as the youngest kid entering kindergarten

EVENTS

There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (

Jones also reports that this formula allows π be calculated, "... to above 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin."

No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to π. He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π. *VFR

The image of l'Hommedieu's diary from Cajori's History of Mathematical Notations

1959 With the admission of Hawaii as a U S State on Aug. 21, 1959, a new executive order called for the creation of a new U S Flag.

"This is a truly historic occasion because for the second time within a year, a new state has been admitted to the union," Eisenhower said to assembled guests in a White House Cabinet Room ceremony. "It had been a long time since any state had been admitted, so to have this 49th and 50th membership of our Union in such a short space is truly a unique experience."

The new flag's design began as a history project for Robert G. Heft, who was a 17-year-old high school student in Lancaster, Ohio, in 1958.

Heft had an idea that Alaska and Hawaii would one day be states, and he set out to design a 50-star flag.

Using his mother's sewing machine, Heft had spent 12 hours using a yardstick while applying his new design of 100 hand-cut stars on each side of the blue canton of an old 48-star flag.

His teacher, who had given him a "B-" for the project, promised he'd change the grade if his flag was accepted by Congress.

Eisenhower made a personal phone call to the shocked Heft to tell him that his flag design had been accepted.

With Executive Order No. 10834, signed on Aug. 21, 1959, Eisenhower selected Heft's flag out of 1,500 designs that had been submitted for consideration.

Heft's teacher made good on his promise and awarded him the coveted "A."

"I never thought when I designed the flag that it would outlast the 48-star flag," said Heft, who later became a teacher and mayor of Napoleon, Ohio, in a 2007 interview with the Grand Rapids Press in Michigan. "I think of all the things it stood for in the past, the things we've done as a nation that we're proud of. It's not a perfect country, but where else would I like to live?" Heft added in the newspaper interview. He died in 2009. * Frederick N. Rasmussen, The Baltimore Sun

Math teachers might point out to students that this is a very mathematical starfield

The next total solar eclipse after 2017 will be on 8 April 2024. Thereafter the next total solar eclipse is on 30 March 2033. Ref. More Mathematical Astronomical Morsels by Jean Meeus; Willmann-Bell, 2002. *NSEC

BIRTHS

From 1700 until 1718 he worked on a catalog of fixed stars, and from 1672 until 1719 he studied Mars extensively. His most famous astronomical discovery was that the ice caps on Mars are not exactly on the rotational poles of that body. He also recognized (in May 1724) that the corona visible during a solar eclipse belongs to the Sun not to the Moon, and he discovered R Hydrae as a variable star. He also helped with the survey based on the Paris Meridian.

He is also credited for the first observation (1723) of what is usually referred to as Poisson's spot, an observation that was unrecognized until its rediscovery in the early 19th century by Dominique Arago. At the time of Arago's discovery, Poisson's spot gave convincing evidence for the contested wave nature of light.

In mathematics he is most known for obtaining the angle in the rhombic dodecahedron shape in 1712, which is still called the Maraldi angle. *Wik A rhombic face of a dodecahedron has diagonals in the proportion of 2:sqrt(2); making the acute angle appx. 109.5

He is also known for the Szemerédi-Trotter theorem in incidence geometry and the Hajnal-Szemerédi theorem in graph theory. Ajtai and Szemerédi proved the corners theorem, an important step toward higher dimensional generalizations of the Szemerédi theorem. With Ajtai and Komlós he proved the ct

DEATHS

Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who wrote about it in 1744 and 1746. Maupertuis felt that "Nature is thrifty in all its actions", and applied the principle broadly. Johann Bernoulli instructed both König and Pierre Louis Maupertuis as pupils during the same period. Konig is also remembered as a tutor to Émilie du Châtelet, one of the few female physicists of the 18th century. *Wik

**1560**The occurrence at the predicted time of a solar eclipse in Copenhagen turned Tycho Brahe toward a life of observational astronomy. *VFR Thony Christie has stated that it was the failure to occur at the predicted time that inspired Tycho. At any rate, he would be able to predict them himself within a few years: A total lunar eclipse occurred on December 8, 1573. It was predicted and then observed by a young Tycho Brahe (assisted by his sister Sophia) at Knutstorp Castle. He said "I cannot but be very surprised that even at this youthful age of 26 years, I was able to get such accurate results." *Wik**1609**Galileo demonstrates his telescope to the aristocrats of Venice. *Renaissance Mathematicus,**1706**Jakob Hermann writes to Leibniz about proof that Machin's series converges to pi. In 1706 William Jones published a work Synopsis palmariorum matheseos or, A New Introduction to the Mathematics, Containing the Principles of Arithmetic and Geometry Demonstrated in a Short and Easie Method ... Designed for ... Beginners. (This is the book in which Jones first uses Pi in the mathematical sense it is now used) This contains on page 243 the following passage:-There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (

^{16}/_{5}-^{4}/_{239}) -^{1}/_{3}(^{16}/_{5}^{3}-^{4}/_{239}^{3}) &c. = 3.14159 &c. = π. This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art. 64.38 may be examined with all desirable ease and dispatch.Jones also reports that this formula allows π be calculated, "... to above 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin."

No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to π. He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π. *VFR

**1776**First recorded use of dollar symbol $. Ezra l'Hommedieu, a member of the New York Provencial Assembly had over a dozen different symbols in his diary beginning with a single vertical bar and proceeding to two vertical bars. *F Carjori (Notes on the term "dollar" and the symbol here.)The image of l'Hommedieu's diary from Cajori's History of Mathematical Notations

**1888**William Seward Burroughs of St. Louis obtained a patent for his adding machine, the ﬁrst successfully marketed. In January, 1886, he incorporated as the American Arithmometer Corporation. *VFR He received patents on four adding machine applications (No. 388,116-388,119), the first U.S. patents for a "Calculating-Machine" that the inventor would continue to improve and successfully market. One year after making his first patent application on 10 Jan 1885, he incorporated his business as the American Arithmometer Corporation of St. Louis, in Jan 1886, with an authorized capitalization of . After Burrough's early death in 1898, after moving from St. Louis to Detroit, Michigan, that company reorganized as the Burroughs Adding Machine Co., incorporated in Jan 1905, with a capital of . The new name was in tribute to the inventor.*TISRecords of the Patent and Trademark Office, 1836 - 1978 |

**1893**The zeroeth International Mathematical Congress with representatives of seven countries was held in conjunction with the Chicago World’s Fair on August 21–25. William E. Story of Clark University was president of the Congress. Felix Klein of Germany came at Kaiser Wilhelm’s personal request. Klein brought nearly all of the mathematical papers published by his countrymen and a superb collection of mathematic models. [AMS Semicentennial Publishers, vol 1, p. 74]. *VFR I have read that Klein's models led to more frequent use of them in American Education.Felix Klein's curious collection of geometric wonders, displayed by Goettingen mathematics department *phalpern |

**1945**Harry Daghlian is exposed to radiation after an accident late in the evening while "Tickling the dragon's tail". This was a coined term for the criticality experiments to determine the amount of fissionable material needed for a sustained chain reaction. He died 25 days later at the hospital in Los Alamos. *atomicheritage.org**1949**John Mauchly and J. Presper Eckert, Jr. demonstrate BINAC, a computer capable of calculating 12,000 times faster than a human being.*VFR (*I wonder how they decided how fast a human being could calculate?*)1959 With the admission of Hawaii as a U S State on Aug. 21, 1959, a new executive order called for the creation of a new U S Flag.

"This is a truly historic occasion because for the second time within a year, a new state has been admitted to the union," Eisenhower said to assembled guests in a White House Cabinet Room ceremony. "It had been a long time since any state had been admitted, so to have this 49th and 50th membership of our Union in such a short space is truly a unique experience."

The new flag's design began as a history project for Robert G. Heft, who was a 17-year-old high school student in Lancaster, Ohio, in 1958.

Heft had an idea that Alaska and Hawaii would one day be states, and he set out to design a 50-star flag.

Using his mother's sewing machine, Heft had spent 12 hours using a yardstick while applying his new design of 100 hand-cut stars on each side of the blue canton of an old 48-star flag.

His teacher, who had given him a "B-" for the project, promised he'd change the grade if his flag was accepted by Congress.

Eisenhower made a personal phone call to the shocked Heft to tell him that his flag design had been accepted.

With Executive Order No. 10834, signed on Aug. 21, 1959, Eisenhower selected Heft's flag out of 1,500 designs that had been submitted for consideration.

Heft's teacher made good on his promise and awarded him the coveted "A."

"I never thought when I designed the flag that it would outlast the 48-star flag," said Heft, who later became a teacher and mayor of Napoleon, Ohio, in a 2007 interview with the Grand Rapids Press in Michigan. "I think of all the things it stood for in the past, the things we've done as a nation that we're proud of. It's not a perfect country, but where else would I like to live?" Heft added in the newspaper interview. He died in 2009. * Frederick N. Rasmussen, The Baltimore Sun

Math teachers might point out to students that this is a very mathematical starfield

**1972**Peru issued a Air Post Stamp picturing a Quipu. [Scott #C341]. *VFR**2015**The AAS Division For Planetary Sciences announced Dr. Dan Durda (Southwest Research Institute) as winner of the Carl Sagan Medal for outstanding communication by an active planetary scientist to the general public, Over 30 years ago in my first full year of teaching, Dan was one of the first of the many bright, kind, and conscientious students who made my years in the classroom wonderful. Congratulations to a great scientist, and scientific communicator.**2017**Next total solar eclipse in the USA. The southern part of Illinois will have 2 total solar eclipses in a time span of only 7 years. Maximum duration will be occur near Hopkinsville, Ky. It will last two minutes and 40 seconds.The next total solar eclipse after 2017 will be on 8 April 2024. Thereafter the next total solar eclipse is on 30 March 2033. Ref. More Mathematical Astronomical Morsels by Jean Meeus; Willmann-Bell, 2002. *NSEC

**1665 Giacomo Filippo Maraldi**(August 21, 1665 – December 1, 1729) was a French-Italian astronomer and mathematician. His name is also given as Jacques Philippe Maraldi. Born in Perinaldo (modern Liguria) he was the nephew of Giovanni Cassini, and worked most of his life at the Paris Observatory (1687 – 1718). He also is the uncle of Jean-Dominique Maraldi.From 1700 until 1718 he worked on a catalog of fixed stars, and from 1672 until 1719 he studied Mars extensively. His most famous astronomical discovery was that the ice caps on Mars are not exactly on the rotational poles of that body. He also recognized (in May 1724) that the corona visible during a solar eclipse belongs to the Sun not to the Moon, and he discovered R Hydrae as a variable star. He also helped with the survey based on the Paris Meridian.

He is also credited for the first observation (1723) of what is usually referred to as Poisson's spot, an observation that was unrecognized until its rediscovery in the early 19th century by Dominique Arago. At the time of Arago's discovery, Poisson's spot gave convincing evidence for the contested wave nature of light.

In mathematics he is most known for obtaining the angle in the rhombic dodecahedron shape in 1712, which is still called the Maraldi angle. *Wik A rhombic face of a dodecahedron has diagonals in the proportion of 2:sqrt(2); making the acute angle appx. 109.5

^{o}. This is also the angle between two segments from the center to the vertices of a tetrahedron. Four soap bubbles intersect at this same angle according to Joseph Plateau's work, and Kepler noticed the shape at the closed ends of honeycombs.*PB NOTES**1757 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**1789 Augustin-Louis Cauchy**(21 Aug 1789;23 May 1857) French mathematician who pioneered in analysis and the theory of substitution groups (groups whose elements are ordered sequences of a set of things). He was one of the greatest of modern mathematicians. *TIS**1901 Edward Copson**(21 Aug 1901; 16 Feb 1980) English mathematician known for his studies in classical analysis, differential and integral equations, and their use in mathematical physics. After graduating from Oxford University with a B.A. degree in 1922, he moved to Scotland where he spent the nearly all of his career. His first book, The Theory of Functions of a Complex Variable (1935) was immediately successful. He was a co-author for his next book, The Mathematical Theory of Huygens' Principle (1939). By 1975, he had published four more books, on asymptotic expansions, metric spaces and partial differential equations. Many of the papers he wrote bridged mathematics and physics, of which his last showed his interest in astrophysics, Electrostatics in a Gravitational Field (1978) which was relevant to Black Holes.*TIS**1932 Louis de Branges de Bourcia**(born August 21, 1932) is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges' theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis.*SAU**1940 Endre Szemerédi**(August 21, 1940, ) is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986. He received his PhD from Moscow State University. His adviser was the late mathematician Israel Gelfand. He has published over 200 scientific articles in the fields of Discrete Mathematics, Theoretical Computer Science, Arithmetic Combinatorics and Discrete Geometry. He is best known for his proof from 1975 of an old conjecture of Paul Erdős and Paul Turán: if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions. This is now known as Szemerédi's theorem. One of the key tools introduced in his proof is now known as the Szemerédi regularity lemma, which has become a very important tool in combinatorics, being used for instance in property testing for graphs and in the theory of graph limits.He is also known for the Szemerédi-Trotter theorem in incidence geometry and the Hajnal-Szemerédi theorem in graph theory. Ajtai and Szemerédi proved the corners theorem, an important step toward higher dimensional generalizations of the Szemerédi theorem. With Ajtai and Komlós he proved the ct

^{2}/log t upper bound for the Ramsey number R(3,t), and constructed a sorting network of optimal depth. With Ajtai, Chvátal, and M. M. Newborn, Szemerédi proved the famous Crossing Lemma, that a graph with n vertices and m edges, where m greater than 4n has at least m^{3}/ 64n^{2}crossings. With Paul Erdős, he proved the Erdős-Szemerédi theorem on the number of sums and products in a finite set. With Wolfgang Paul, Nick Pippenger, and William Trotter, he established a separation between nondeterministic linear time and deterministic linear time, in the spirit of the infamous P versus NP problem. With William Trotter, he established the Szemerédi–Trotter theorem obtaining an optimal bound on the number of incidences between finite collections of points and lines in the plane.*Wik**1757 Samuel König**(July 31, 1712, Büdingen – August 21, 1757, Zuilenstein near Amerongen) was a German mathematician who is best remembered for his part in a dispute with Euler over the Principle of Least Action.*SAU In the 17th century Pierre de Fermat postulated that "light travels between two given points along the path of shortest time," which is known as the principle of least time or Fermat's principle.Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who wrote about it in 1744 and 1746. Maupertuis felt that "Nature is thrifty in all its actions", and applied the principle broadly. Johann Bernoulli instructed both König and Pierre Louis Maupertuis as pupils during the same period. Konig is also remembered as a tutor to Émilie du Châtelet, one of the few female physicists of the 18th century. *Wik

**1762 Lady Mary Wortley Montagu**(15 May 1689 – 21 August 1762) was an English aristocrat, writer, and poet. She is listed in this blog because during her life in Turkey with her diplomat husband, she realized that the Turkish people almost never suffered the scars of smallpox. Investigating, she found that mothers would take a scab from a smallpox sufferer and open their children's veins in several places and plant the scabs there. She did the same to her own children and when she returned to England she shared the idea with her friends and tried to make its use more broad. She persuaded hte Princess of Wales to test the treatment and seven prisoners' awaiting execution were offered the treatment in exchange for clemency. All survived, and were released. Catherine the Great and her son Paul were both inoculated. When Lady Mary Montagu died in 21 August of 1762, Edward Jenner was thirteen years old. HT @drSueMoss

**1814 Count Benjamin Thompson Rumford**(26 Mar 1753, 21 Aug 1814) American-born British physicist, government administrator, and a founder of the Royal Institution of Great Britain, London. Because he was a Redcoat officer and an English spy during the American revolution, he moved into exile in England. Through his investigations of heat he became one of the first scientists to declare that heat is a form of motion rather than a material substance, as was popularly believed until the mid-19th century. Among his numerous scientific contributions are the development of a calorimeter and a photometer. He invented a double boiler, a kitchen stove and a drip coffee pot. *TIS

**1836 Claude-Louis Navier**(10 February 1785 – 21 August 1836) was a French mathematician best known for the Navier-Stokes equations describing the behaviour of a incompressible fluid. *SAU Navier also formulated the general theory of elasticity in a mathematically usable form (1821), making it available to the field of construction with sufficient accuracy for the first time. In 1819 he succeeded in determining the zero line of mechanical stress, finally correcting Galileo Galilei's incorrect results, and in 1826 he established the elastic modulus as a property of materials independent of the second moment of area. Navier is therefore often considered to be the founder of modern structural analysis. *Wik

**1927 William Burnside**(2 July 1852 – 21 August 1927) wrote the first treatise on groups in English and was the first to develop the theory of groups from a modern abstract point of view. *SAU

Burnside is also remembered for the formulation of Burnside's problem (which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it) and for Burnside's lemma (a formula relating the number of orbits of a permutation group acting on a set with the number of fixed points of each of its elements) though the latter had been discovered earlier and independently by Frobenius and Cauchy.

In addition to his mathematical work, Burnside was a noted rower; while he was a lecturer at Cambridge he also coached the crew team. In fact, his obituary in The Times took more interest in his athletic career, calling him "one of the best known Cambridge athletes of his day". *Wik

**1957 Harald Ulrik Sverdrup**( 15 Nov 1888; 21 Aug 1957)was a Norwegian meteorologist and oceanographer known for his studies of the physics, chemistry, and biology of the oceans. He explained the equatorial countercurrents and helped develop the method of predicting surf and breakers. As scientific director of Roald Amundsen's polar expedition on Maud (1918-1925), Sverdrup worked extensively on meteorology, magnetics, atmospheric electricity, physical oceanography, and tidal dynamics on the Siberian shelf, and even on the anthropology of Chukchi natives. In 1953, Sverdrup quantified the concept of "critical depth", explaining the onset of the spring phytoplankton bloom in newly stratified water columns.*TIS

**1995 Subrahmanyan Chandrasekhar**(19 Oct 1910, 21 Aug 1995) Indian-born U.S. astrophysicist who shared with William A. Fowler the 1983 Nobel Prize for Physics for formulating the currently accepted theory on the later evolutionary stages of massive stars, work that subsequently led to the discovery of neutron stars and black holes. *TIS

**2012 William Paul Thurston**(October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. He was last a professor of mathematics and computer science at Cornell University (since 2003). *Wik His AMS obituary begins:

William P. Thurston, whose geometric vision revolutionized topology, died August 21 at the age of 65. Within a short span of just a few years at the beginning of his career, Thurston proved so many outstanding results in foliation theory, that the whole area seemed to be finished because he had answered most of the important open problems. Then, in the mid-1970s, he turned his attention to low-dimensional topology, to which he brought a whole new set of geometric tools, most notably from hyperbolic geometry.

** yes, but only one, 1597

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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