3-5, 5-7, 11-13, 17-19... 29-31
Pat'sBlog
The mathematical (and other) thoughts of a (now retired) math teacher,
Saturday, 18 July 2026
Chen, Goldbach, and the Search for an Unsolved Proof
On This Day in Math - July 18
-Danica McKellar
(I can't believe I'm doing math quotes by "Winnie" from Wonder Years)
The 199th day of the year; 199 is prime (in fact, all three permutations of the number are prime) and is the sum of three consecutive primes: 61 + 67 + 71, and of five consecutive primes: 31 + 37 + 41 + 43 + 47. (Suddenly struck me I don't know what is the smallest prime that is the sum of consecutive primes in more than one way!)(So the answer was right in front of my face, one of the primes listed above)
199 is the smallest number with an additive persistence of 3. (iterate the sum of the digits. The number of additions required to obtain a single digit from a number n is called the additive persistence of n, and the digit obtained is called the digital root of n. ) 1+9+9 =19, 1+9=10, 1+0 = 1. so the additive persistence is 3 and the digital root is 1.
I like "almost constants". For the 199th day,\( ( \frac{\sqrt{5} +1}{2})^{11}= 199.0050249987406414902082… \)
199 is the last year day that is part of a prime quadruplet, (191, 193, 197, 199)
199 is the smallest number that has an additive persistence of 3, 1+9+9 = 19; 1+9 =10; 1+0=3 *Prime Curios
199 = 100^2-99^2
199 as a palindrome of its own digits, 99+1+99=199= 9*9+9*1+9+1+9+1*9+9*9
199 is a permutable prime, and 919 and 991 are both prime
199 is the first prime number in a sequence of 10 consecutive prime numbers with common difference 210 (tao and green 2008; see R.Taschner "Die Farben der Quadratzahlen" p. 147)the ten primes are 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, and 2089.
The next prime after 199 is 211. If they are concatenated in either order, they form a prime, 199211 and 211199 are both prime. *Prime Curios
199 is the smallest emirp (991 is prime also) that is also an invertible prime, it's 180 degree rotation (Strobagram) produces the prime 661. *Prime Curios
199, 211, and 223 are the smallest triple of primes of the form n, n+12 and n+24, and it is the only triple less than 1000. *Prime Curios
go here more Math Facts for every Year Date,
1765 The Board of Longitude appointed Richard Dunthorne to be the first "Comparer of the Ephemeris and Corrector of the Proofs" for the (then still future) Nautical Almanac and Astronomical Ephemeris. *Wik Later there would be a small team of these "computers" creating lunar tables as a potential solution for the "longitude problem", determining longitude at sea. At this time calculator was a term used occasionally for accountants, but more commonly for a book of mathematical tables.One example of such was "The Assistant Calculator, or Cotton Spinners Guide, being a complete set of tables, of the greatest use in the cotton spinning business."
Dunthorne was born in humble circumstances in Ramsey, Cambridgeshire, where he attended the free grammar school. There he attracted the notice of Roger Long (later Master of Pembroke Hall, Cambridge), whose protégé Dunthorne became. Dunthorne moved to Cambridge where Long first appointed him as a "footboy", and where he received some further education (though this does not seem to have been regular university education). Dunthorne then "managed" a preparatory school in Coggeshall, Essex, and later returned to Cambridge where Long obtained for him an appointment as a "butler" at Pembroke Hall, an office that Dunthorne retained for the rest of his life. Here, Dunthorne's main activity seems to have been in assisting Long in astronomical and scientific work.
Dunthorne also held an appointment for some years, concurrently with his work with Long, as superintendent of works of the Bedford Level Corporation, responsible for water management in the Fens; he began this work several years" before 1761, continuing into the 1770s. In this role, Dunthorne was concerned in a survey of the Fens in Cambridgeshire, and he also supervised construction of locks near Chesterton on the River Cam.
Dunthorne's association with Long remained lifelong, and in the end Dunthorne acted as executor of Long's will. *Wik
1860 First wet plate photographs of an eclipse; they require 1/30 of the exposure time of a daguerreotype. *NSEC
Photo :Stanford SOLAR Center - History of Solar
1872 Weierstrass, in a lecture to the Berlin Academy, gave his classical example of a continuous nowhere differentiable function. See Big Kline, p. 956.*VFR
1879 On this day in 1879, George Chrystal was appointed to the chair of mathematics in the University of Edinburg. He is primarily known for his books on algebra and his studies of seiches. [a temporary disturbance or oscillation in the water level of a lake or partially enclosed body of water, especially one caused by changes in atmospheric pressure.]
He was educated at Aberdeen Grammar School and the University of Aberdeen. In 1872, he moved to study under James Clerk Maxwell at Peterhouse, Cambridge. He graduated Second Wrangler in 1875, joint with William Burnside, and was elected a fellow of Corpus Christi. He was appointed to the Regius Chair of Mathematics at the University of St Andrews in 1877, and then in 1879 to the Chair in Mathematics at the University of Edinburgh. In 1911, he was awarded the Royal Medal of the Royal Society for his researches into seiches (the surface oscillations of Scottish lochs).
1898 Marie and Pierre Curie discover the previously unknown element Polonium which she named for her home country, Poland. *Brody & Brody, The Science Class You Wished You Had
1962 Hearings on Mercury 13 Women suspended. The first potential US women in space, often called the Mercury 13 in comparison to the original Mercury 7 astronauts, had a hearing in congress beginning July 17th. The house convened public hearings before a special Subcommittee on Science and Astronautics. Significantly, the hearings investigated the possibility of gender discrimination two full years before the Civil Rights Act of 1964 made that illegal, making these hearings a marker of how ideas about women's rights permeated political discourse even before they were enshrined in law. The hearings would abruptly be terminated at lunch on the 18th. In less than a year, Soviet cosmonaut Valentina Tereshkova became the first woman in space on June 16, 1963. In response, Clare Boothe Luce published an article in Life criticizing NASA and American decision makers. By including photographs of all thirteen Lovelace finalists, she made the names of all thirteen women public for the first time. (The Time issue is available at Google Books here. Astronaut Sally Ride became the first American woman in space in 1983 on STS-7. *Wik
1968 Intel Founded. Robert Noyce, Andy Grove and Gordon Moore incorporated Intel, a company they built on production of the microprocessor. The component that has allowed computers to increase in speed and decrease in size, the microprocessor also built Intel, whose Pentium processors now power most IBM-compatible personal computers.
Moore is famous for Moore's Law, which dictates that every 18 months microprocessors double in speed and decrease in size by half.
1979 Great Britain issued a stamp honoring Alice’s Adventures in Wonderland. *VFR
2014 first "Sun-spotless day" on the Earthward side of sun since 2011, *David Dickinson @Astroguyz
Spaceweather.com reports that today we surpassed the largest number of spotless days (270) of the previous 2008 Solar Minimum cycle. The current spotless streak stands at 33 days and is quite possibly on its way to surpass the previous longest streak of this minimum at 36 days. And you have to go back to 1913 to find a year that had more spotless days (311)!
You might be wondering: when is the next Solar Maximum? That’s forecast to be July 2025. Both the minimum & maximum forecasts have a +/- 6-month error. *The Swinging Post
The blank sun on Dec. 8, 2019. Credit: NASA/Solar Dynamics Observatory
1013 Hermann of Reichenau (July 18, 1013 – September 24, 1054), was a German mathematician who was important for the transmission of Arabic mathematics, astronomy and scientific instruments into central Europe.*SAU
Blessed Hermann of Reichenau or Herman the Cripple (18 July 1013 – 24 September 1054), also known by other names, was an 11th-century Benedictine monk and scholar. He composed works on history, music theory, mathematics, and astronomy, as well as many hymns. He has traditionally been credited with the composition of "Salve Regina", "Veni Sancte Spiritus", and "Alma Redemptoris Mater", although these attributions are sometimes questioned. His cultus and beatification were confirmed by the Roman Catholic Church in 1863. *Wik
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1635 Robert Hooke ( 18 July[NS 28 July] 1635 – 3 March 1703) born.English natural philosopher, architect and polymath. His adult life comprised three distinct periods: as a scientific inquirer lacking money; achieving great wealth and standing through his reputation for hard work and scrupulous honesty following the great fire of 1666, but eventually becoming ill and party to jealous intellectual disputes. These issues may have contributed to his relative historical obscurity.
He was at one time simultaneously the curator of experiments of the Royal Society and a member of its council, Gresham Professor of Geometry and a Surveyor to the City of London after the Great Fire of London , in which capacity he appears to have performed more than half of all the surveys after the fire. He was also an important architect of his time, though few of his buildings now survive and some of those are generally misattributed, and was instrumental in devising a set of planning controls for London whose influence remains today. Allan Chapman has characterised him as "England's Leonardo" *wik
He was born in Freshwater, Isle of Wight, and discovered the law of elasticity, known as Hooke's law, and invented the balance spring for clocks. He was a virtuoso scientist whose scope of research ranged widely, including physics, astronomy, chemistry, biology, geology, architecture and naval technology. On 5 Nov 1662, Hooke was appointed the Curator of Experiments at the Royal Society, London. After the Great Fire of London (1666), he served as Chief Surveyor and helped rebuild the city. He also invented or improved meteorological instruments such as the barometer, anemometer, and hygrometer. Hooke authored the influential Micrographia (1665)*TIS
One of my favorites, a louse seems to be marching off to war
1768 Jean Robert Argand born (July 18, 1768 – August 13, 1822). His single original contribution to mathematics was the invention and elaboration of a geometric representation of complex numbers and operations on them. In this he was preceded by Wessel and followed by Gauss.*VFR Swiss mathematician who was one of the earliest to use complex numbers, which he applied to show that all algebraic equations have roots. He invented the Argand diagram - a geometrical representation of complex numbers as a point with the real portion of the number on the x axis and the imaginary part on the y axis.*Wik
1813 Pierre Laurent (July 18, 1813 – September 2, 1854) was a French mathematician best-known for his study of the so-called Laurent Series in Complex analysis. *SAU
The Laurent series is an expansion of a function into an infinite power series, generalizing the Taylor series expansion.*Wik
1853 Antoon Lorentz (18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the transformation equations subsequently used by Albert Einstein to describe space and time. *Wik
Lorentz is best known for his work on electromagnetic radiation and the FitzGerald-Lorentz contraction. He developed the mathematical theory of the electron.*SAU
1856 Giacinto Morera (Novara, 18 July 1856 – Turin, 8 February 1909), was an Italian engineer and mathematician. He is remembered for Morera's theorem in the theory of functions of a complex variables and for his work in the theory of linear elasticity.
Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies{\displaystyle \oint _{\gamma }f(z)\,dz=0}
for every closed piecewise C1 curve 𝛾{\displaystyle \gamma } in D must be holomorphic on D.
He was member of the Accademia Nazionale dei Lincei (first elected corresponding member on 18 July 1896, then elected national member on 26 August 1907)[20] and of the Accademia delle Scienze di Torino (elected on 9 February 1902).[21] Maggi (1910, p. 317) refers that also the Kharkov Mathematical Society elected him corresponding member during the meeting of the society held on 31 October 1909 (Old Calendar), being apparently not aware of his death.*Wik
1891 Emil Julius Gumbel (18 July 1891, in Munich – 10 September 1966, in New York City) was an American mathematician and statistician is known for his work in reliability theory and order statistics. His name remains on the Type 1 extreme value distribution, known as the Gumbel distribution. In his early life, during WW I, he militantly advocated for pacifism. Gumbel acted as a historian and statistician recording the political murders in the early Weimar Republic. His activities caused ostracization and political harassment. When he moved to France (1932), he could better concentrate on his mathematical work, most notably examining the statistical distributions most used by actuaries. He also applied his skills in the fields of hydrology (floods) and meteorology (drought). Eight years later, due to WW II, he moved to the U.S. (1940) and continue this work. *TiS
1899 Robert Schlapp (18 July 1899 in Edinburgh, Scotland - 31 May 1991 in Ashford, Kent, England)studied at Edinburgh and Cambridge universities. He spent his whole career at Edinburgh University teaching mathematics and Physics. He was also interested in the History of Mathematics. He became President of the EMS in 1942 and 1943. *SAU
He was born in Edinburgh on 18 July 1899, the youngest of three children of Anna Lotze and Otto Schlapp.[2] His father only appears in Post Office Directories around 1910, at which point he is listed as a university lecturer living at 54a George Square. His father lectured in German at the University of Edinburgh and later (1926) became the University's first Professor of German.
In the First World War, obviously a potential problem due to his German background, he enlisted under the Derby Scheme and joined the 31st battalion of the Middlesex Regiment in 1917 at the age of 18. This was a labouring unit rather than a fighting battalion, involved in tasks such as trench construction. After the war, Schlapp studied mathematics and physics at the University of Edinburgh graduating MA around 1923 then doing postgraduate studies at the University of Cambridge gaining a doctorate (PhD) in 1925.
Returning to the University of Edinburgh he began lecturing in Natural Philosophy (Physics) and Applied Mathematics in autumn 1925. He became Senior Lecturer in Mathematical Physics in 1927.[6] In this role he was assistant to Charles Galton Darwin (who had recently replaced Cargill Gilston Knott).
In 1927, he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, Sir Charles Galton Darwin, David Gibb, and Edward Thomas Copson. He served as Curator of the Society's artefacts from 1959 to 1969 and as their Vice President from 1969 to 1972.
In 1983, he won the Society's bicentenary medal (presented to him by Queen Elizabeth II). He was President of the Edinburgh Mathematical Society.
In 1936 Professor Darwin retired and was replaced by Max Born whom Schlapp then assisted in turn. Schlapp retired in 1969, and died in Ashford in Kent on 31 May 1991.
Whilst (A beautiful British word) playing cello in his brother, Walter Schlapp's, string quartet, he met Mary Fleure (who played second violin). He married her in 1940. They had two daughters. Mary died in 1975.
1906 Edwin Ford Beckenbach (July 18, 1906 – September 5, 1982) was an American mathematician
In 1929 he earned a master's degree at Rice University, and in 1931 a PhD under the direction of Lester R. Ford. As a postdoc, he was a National Research Fellow at Princeton University, Ohio State University, and the University of Chicago.
At UCLA, he led the development of the graduate program in mathematics. The first mathematics PhD was granted under his direction as thesis advisor.
Beckenbach was also a leader in the founding (in 1948) of the Institute of Numerical Analysis, which was then a branch of the National Bureau of Standards. His institute developed in 1948 and 1949 a vacuum-tube computer (SWAC), which began operation in July 1950 and was for a short time the fastest computer in the world. In 1974 he retired from UCLA as professor emeritus. From 1949 to 1963, he was a consultant for the Rand Corporation and in the academic year 1951/1952 he was a visiting professor at the Institute for Advanced Study.
In the academic year 1958/59 he was a Guggenheim Fellow at ETH Zürich. With František Wolf, Beckenbach founded in 1951 the Pacific Journal of Mathematics, of which he was the first editor. In 1983, he received the Distinguished Service Award from the Mathematical Association of America. The Beckenbach Book Prize, first awarded in January 1985, is named in his honor. *Wik
1922 Thomas S(amuel) Kuhn (July 18, 1922 – June 17, 1996) was an American historian of science, MIT professor, noted for The Structure of Scientific Revolutions (1962), one of the most influential works of history and philosophy written in the 20th century. His thesis was that science was not a steady, cumulative acquisition of knowledge, but it is "a series of peaceful interludes punctuated by intellectually violent revolutions." Then appears a Lavoisier or an Einstein, often a young scientist not indoctrinated in the accepted theories, to sweep the old paradigm away. Such revolutions, he said, came only after long periods of tradition-bound normal science. "Frameworks must be lived with and explored before they can be broken," *TIS This was the first modern use of the term "paradigm" in this way.
1939 Marjorie Lee Senechal (née Wikler,July 18,1939 - ) is an American mathematician and historian of science, the Louise Wolff Kahn Professor Emerita in Mathematics and History of Science and Technology at Smith College and editor-in-chief of The Mathematical Intelligencer. In mathematics, she is known for her work on tessellations and quasicrystals; she has also studied ancient Parthian electric batteries and published several books about silk.
Senechal won the Mathematical Association of America's Carl B. Allendoerfer Award for excellence in expository writing in Mathematics Magazine in 1982, for her article, "Which Tetrahedra Fill Space?" In 2008, her book American Silk 1830 – 1930 won the Millia Davenport Publication Award of the Costume Society of America. In 2012, she became a fellow of the American Mathematical Society.
1948 Michel Hartmut (German pronunciation: [18 July, 1948 - ) is a German biochemist, who received the 1988 Nobel Prize,along with Johann Deisenhofer and Robert Huber, in Chemistry for determination of the first crystal structure of an integral membrane protein, a membrane-bound complex of proteins and co-factors that is essential to photosynthesis.
They are the first to succeed in unraveling the full details of how a membrane-bound protein is built up, revealing the structure of the molecule atom by atom. The protein is taken from a bacterium which, like green plants and algae, uses light energy from the sun to build organic substances. All our nourishment has its origin in this process, which is called photosynthesis and which is a condition for all life on earth.*TiS
1650 Christoph Scheiner SJ (25 July 1573 (or 1575) – 18 July 1650) was a Jesuit priest, physicist and astronomer in Ingolstadt. In 1603, Scheiner invented the pantograph, an instrument which could duplicate plans and drawings to an adjustable scale. Later in life he would invent a sunspot viewing appartus. In 1611, Scheiner observed sunspots; in 1612 he published the "Apelles letters" in Augsburg. Marcus Welser had the first three Apelles letters printed in Augsburg on January 5, 1612. They provided one of many reasons for the subsequent unpleasant argument between Scheiner and Galileo Galilei. *Wik Thus, in 1614, Galileo found himself in an unresoved dispute over priority with a mean and determined Jesuit. The fight was to grow meaner in subsequent years. It would play a major role in Galileo's Inquisitional trial eighteen years later. *James Reston, Jr., Galileo: A Life
1742 Abraham Sharp (1653– 18 July 1742) was an English mathematician who worked with Flamsteed. He calculated π to 72 places (using an arcsine sequence, briefly holding the record until John Machin calculated 100 digits in 1706).*SAU
1807 Thomas Jones (23 June 1756 – 18 July 1807) was Head Tutor at Trinity College, Cambridge for twenty years and an outstanding teacher of mathematics. He is notable as a mentor of Adam Sedgwick.
He was born at Berriew, Montgomeryshire, in Wales. On completing his studies at Shrewsbury School, Jones was admitted to St John's College, Cambridge on 28 May 1774, as a 'pensioner' (i.e. a fee-paying student, as opposed to a scholar or sizar). He was believed to be an illegitimate son of Mr Owen Owen, of Tyncoed, and his housekeeper, who afterwards married a Mr Jones, of Traffin, County Kerry, Thomas then being brought up as his son.
On 27 June 1776, Jones migrated from St John's College to Trinity College. He became a scholar in 1777 and obtained his BA in 1779, winning the First Smith's Prize and becoming Senior Wrangler. In 1782, he obtained his MA and became a Fellow of Trinity College in 1781. He became a Junior Dean, 1787–1789 and a Tutor, 1787-1807. He was ordained a deacon at the Peterborough parish on 18 June 1780. Then he was ordained priest, at the Ely parish on 6 June 1784, canon of Fen Ditton, Cambridgeshire, in 1784, and then canon of Swaffham Prior, also 1784. On 11 December 1791, he preached before the University, at Great St Mary's, a sermon against duelling (from Exodus XX. 13), which was prompted by a duel that had lately taken place near Newmarket between Henry Applewhaite and Richard Ryecroft, undergraduates of Pembroke, in which the latter was fatally wounded. Jones died on 18 July 1807, in lodgings in Edgware Road, London. He is buried in the cemetery of Dulwich College. A bust and a memorial tablet are in the ante-chapel of Trinity College. *Wik
1930 Karl Emmanuel Robert Fricke (September 24, 1861 in Helmstedt, Germany ; July 18, 1930 in Bad Harzburg, Germany) was a German mathematician, known for his work in function theory, especially on elliptic, modular and automorphic functions. He was one of the main collaborators of Felix Klein, with whom he produced two classic two volume monographs on elliptic modular functions and automorphic functions.
In 1893 in Chicago, his paper Die Theorie der automorphen Functionen und die Arithmetik was read (but not by Fricke) at the International Mathematical Congress held in connection with the World's Columbian Exposition. From 1894 to 1930 Fricke was professor of Higher Mathematics at the Technische Hochschule Carolo-Wilhelmina in Braunschweig.*Wik
1977 Georgi Delchev Bradistilov (25 October 1904 [12 October 1904 O.S.] – 18 June 1977) was a Bulgarian mathematician.
He attended 3rd Sofia gymnasium and in 1922 entered Sofia University to study physics and mathematics. In 1927 he graduated with honors and the same year was appointed as assistant professor in mathematics. In the 1930s he studied at the University of Paris and the University of Munich. Bradistilov was one of the last students to take Arnold Sommerfeld's course in theoretical physics before his retirement. In 1938, he defended his doctorate, with Oskar Perron as advisor, at the University of Munich.
Georgi Bradistilov's contributions to applied mathematics are related to nonlinear differential equations and their applications to mechanics and electrotechnics, to electrostatic potential, to nonlinear oscillations.
He was notorious for his sense of humor and openness, for his love of arts and nature as well as for his refined taste, his wife being an artist educated in Florence.(QED?)
During his lifetime Georgi Bradistilov received many Bulgarian state decorations and awards. Recently a street in Sofia near the Technical University was named after him. *Wik
2018 Burton Richter (22 Mar 1931, ) American physicist who was jointly awarded the 1976 Nobel Prize for Physics with Samuel C.C. Ting for the discovery of a new subatomic particle, the J/psi particle. *TIS He led the Stanford Linear Accelerator Center (SLAC) team which co-discovered the J/ψ meson in 1974, alongside the Brookhaven National Laboratory (BNL) team led by Samuel Ting. This discovery was part of the so-called November Revolution of particle physics. He was the SLAC director from 1984 to 1999.*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
Friday, 17 July 2026
Some History Notes on Casting Out Nines, and an extension you probably never heard of.
I'm Re-posting some old blogs to try and preserve some notes, and do some editing
In my youth, back in the dark ages before calculators, one of the common mathematical tools we were taught was called casting out nines. It seems it is not common anymore, perhaps the errors students make with calculators no longer submit to the simple check of this ancient method. If you happen to be of the generation who have been kissed on the forehead by the Gods of Electronics and actually don't know the method, you can find some notes on my page, and also here is a brief video from YouTube.
I have a note on my MathWords page on the subject from a respected math historian (Albrecht Heefer) that tells me, "Casting out nines is believed to be of Indian origin, but it does not occur before 950. Maximus Planudes called it 'Arithmetic after the Indian method". Along the way I seem to have a note from him telling me that I can find more confirmation on the web site of David Singmaster, the famous historian of mathematical recreation; but while searching there, I seem to have a note that claims the first mention of casting out nines was by the Latin writer Iamblichus in 325 AD... But he was talking about Nichomachus, a Pythagorean who lived around 100 AD.
"325 Iamblichus: On Nicomachus's Introduction to Arithmetic - first mention of Casting Out Nines; first description of the Bloom of Thymarides; first Amicable Numbers."
Now the common thought, or at least as I thought I understood it, was that the inventors of the hindu-arabic numerals had developed casting out nines and it sort of made its way into the west with the introduction of the Arabic numbers. S. S. Gupta credits Aryabhata II, a tenth century Indian mathematician and astronomer as writer of the oldest know surviving work that mentions casting out nines, Mahasiddhanta. Leonardo of Pisa, the famous Fibonacci whose bunny sequence you remember from school (of course you do, 1, 1, 2, 3, 5, 8, 13, 21...... That sequence) was a major influence in bringing both to the west with his famous book, the Liber Abaci, (the book of calculating) around 1202.
But the fact is that the general public held on to their Roman numerals for several centuries, and legal documents had to have them in some areas up into the 15th century.. Now the problem, at least for me, is that it seemed much less likely that someone would develop casting out nines using Roman numerals.. see if you are using Arabic numerals, you take a number and add up the digits... 2534 would give 2+5+3+4 = 14 and then adding 1+4 = 5 so we know that if you divide 2534 by nine, you get a remainder of 5. Now in Roman numerals we write 2534 as MMDXXXIIII ... So I set about trying to figure out casting out nines with Roman numerals, and it hit me.
Anyway, I'm still looking for that Rogue Scholar out there who happens to have the original of Nicomachus' "Introduction to Arithmetic" laying around on his bookshelf and would like to translate for me to explain where he says it came from (if indeed he did).
The reason I am reminded of all this is that I just read an interesting article by the almost unknown English mathematician, Henry Wilbraham (July 25, 1825 – February 13, 1883), in an old Cambridge and Dublin Mathematical Journal. He points out that you can construct a similar division technique for any number. The idea is to use the period of the smaller numbers repeating fraction to break apart the second number. As an example, if you wanted to test to see if some large number was divisible by 37, you would first find the digital period length of the decimal 1/37. It turns out that 1/37 = 0.02702702702702703 so its period is three.
Now we take the really big number we want to test, say 7,424,883,933,621. We want to know if that number is evenly divisible by 37, and if it isn't, what the remainder will be.
The variation in Wilbraham's approach, and as I point out later it's not really a variation at all, is to break the larger number up into sections of three digits (the period of our divisor's reciprocal), so we would add the 621+933+883+424+7= 2868. Now just as we can continue to compute digital roots when casting out nines, because there are more than three digits here, we can recombine those to get 2+868 = 870. Now all we have to do is divide 37 into 870 and if it goes evenly, it's a factor of the larger 13 digit number. If not, the remainder we get will be the same as the remainder when dividing the original number.
Turns out 37 is not a factor of 870 but leaves a remainder of 19. The good news is that we know that when we divide 7,424,883,933,621 by 37, we will get the same remainder.
It turns out that the reason this works is the same as the reason that casting out nines works. The period of 1/9 is one,.11111....., so we add every digit.
The math behind this is simple enough that I think any bright high school kid could understand it. If the period of a numbers reciprocal 1/n is some number p, then it must be true that 10p-1 is divisible by n. In my example, 103-1 must be divisible by 37, and is.
So if we break our larger number, N, up into periods of p, and express the sets of digits as individual numbers, A,B,C,D... so that N= A+10pB + 102p...etc.
So we know that N= A+ (kn+1)B+ (kn+1)2 C.... and if we distribute all these kn+1 terms all the kn powers can be collected (and are thus a multiple of our smaller divisor, n) and the rest will be A+B+C... which is the sum of the periods, and thus the remainder. If this number is longer than the period of 1/n, we can apply it again by using the same reasoning.
I should point out, because Wilbraham is so unknown, he did not spend his entire mathematical life doing arithmetic novelties. He is known for discovering and explaining the Gibbs phenomenon, the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity, nearly fifty years before J. Willard Gibbs did. Gibbs and Maxime Bôcher, as well as nearly everyone else, were unaware of Wilbraham's work on the Gibbs phenomenon.
On This Day in Math - July 17
~Henri Poincaré
The 198th day of the year; 198 is a Harshad number, divisible by the sum of its digits. A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1997.
(Students might try to find a pair of consecutive numbers greater than 10 which are harshad numbers)
198 nines followed by a one is prime 9999...... 91. *Derek Orr@Derektionary
198 is between the twin primes 197 and 199. If you multiply 198 by its reversal, 891, you get 176,418 which is between the twin primes 176,417 and 176,419. Is there another example of this curiosity?
198 = 13^2 + 5^2 + 2^2
Palindrome expressions for 198 = 2 x 72 + 27 x 2 = 3 x 15 + 51 x 3 = 55 + 88+ 55
19 is the 8th prime number, and if you concatenate them, 198 is the (19*8= 152nd) composite number.
More Math Facts for every year date here
In 709 BC, the earliest record of a confirmed total solar eclipse was written in China. From: Ch'un-ch'iu, book I: "Duke Huan, 3rd year, 7th month, day jen-ch'en, the first day (of the month). The Sun was eclipsed and it was total." This is the earliest direct allusion to a complete obscuration of the Sun in any civilization. The recorded date, when reduced to the Julian calendar, agrees exactly with that of a computed solar eclipse. Reference to the same eclipse appears in the Han-shu ('History of the Former Han Dynasty') (Chinese, 1st century AD): "...the eclipse threaded centrally through the Sun; above and below it was yellow." Earlier Chinese writings that refer to an eclipse do so without noting totality.*TIS
In 1778, David Rittenhouse observed a total solar eclipse in Philadelphia. In a letter to him, dated 17 Jul 1778, Thomas Jefferson wrote that "We were much disappointed in Virginia generally on the day of the great eclipse, which proved to be cloudy." Rittenhouse (1732-1796) was not only an American astronomer, but also a mathematician and public official. He is reputed to have built the first American-made telescope and was the first director of the U.S. Mint (1792-1795).*TIS
Jefferson was an excellent applied mathematician and had contacted Rittenhouse on another occasion. Travelling through France ten years later, " in 1788, he noticed peasants near Nancy ploughing, and fell to wondering about the design of the moldboard, that is, the surface which turns the earth: he spent the next ten years working on this, on and off, wondering how to achieve the most efficient design, both offering least frictional resistance, and which also would be easy for farmers out in the frontiers to construct, far from technical help. He consulted the Pennsylvania mathematician Robert Patterson (born in Ireland in 1743), and consulted also another Philadelphia luminary, the self-taught astronomer and mathematical instrument-maker David Rittenhouse (1732-1796)." Jefferson also communicated with Thomas Paine about bridge design, suggesting the use of catenary arches. Jefferson is believed to be the first person ever to use the term "catenary" in English.
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| Charles Wilson Peale's Rittenhouse, *Wik |
1831 This is the date of birth of the man who would produce the birth of the modern slide rule, Victor Mayer Mannheim. The slide rule began with the creation of a single wooden logarithmic scale, the Gunter scale, a few decades after the creation of logarithms by Napier. Within a half dozen years William Oughtred came up with the first paired scales, a pair of linear log scales, and another that were circular. Minor changes happened to the scale for over two hundred years. For example, Peter Mark Roget whose name is associated so closely with the thesaurus, in 1815 added the log-log scale (the logarithm of the logarithm of the number on the C and D scales... ) on the slide rule which facilitated finding powers and roots of numbers. Finally, in 1850, Mannheim would add the sliding runner, the hair line that ties together the various scales. Some suggest Newton may have created a single earlier version when he had a rule made to solve cubics requiring alignments of three log scales, but no written or physical evidence of such has been found.
1850 Vega became the first star (other than the Sun) to be photographed, when it was imaged by William Bond and John Adams Whipple at the Harvard College Observatory. The photo was a daguerreotype.
Astrophotography, the photography of celestial objects, began in 1840 when John William Draper took an image of the Moon using the daguerreotype process. Vega is the brightest star in the northern constellation of Lyra. *Wik
1860 Four years before his death, partly due to his wife, Mary (following the recommendations of a doctor who advocated homeopathic cold water cures) made George Boole lie shivering for hours between cold wet sheets, George Boole wrote a letter to Augustus DeMorgan about his belief in homeopathy. George was a follower of homeopathic practices, but perhaps with a little less enthusiasm than his wife. He had walked home in a cold wet rain and this was the homeopathic treatment... a hair of the dog, so to speak.
1879 It was announced in Nature that Alfred Bray Kempe had proved the four-color conjecture. A correct proof, based on Kempe’s ideas, had to wait another century. [N. L. Biggs, et al., Graph Theory 1736–1936, p. 94] *VFR
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| Alfred Bray Kempe |
1902 A solution for a humidity and heat problem in a printing process became the modern day air conditioner. Willis Carrier, after graduation from Cornell University in 1901 with a Masters of Engineering degree, began work as a research engineer with the Buffalo Forge Company as a research engineer.
1935 The first problem was entered into the Scottish Book, a large bound notebook that Stefan Banach brought to the Scottish Cafe in LLw´ow for mathematicians to record research problems. Many of the problems offered prizes to the solver. They ranged from “2 small beers” to “100 grammes of caviar.” This book has been translated into English and edited by R. D. Mauldin. (below) *VFR ..(A PDF file of the book is now available, thanks to a tip from Robin Whitty at theoremoftheday.org )in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in functional analysis and
topology. Stanislaw Ulam recounts that the tables of the café had marble tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the Scottish Book. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. For problem 153, which was later recognized as being closely related to Stefan Banach 's "basis problem", Stanislaw Mazur offered the prize of a live goose. This problem was solved only in 1972 by Per Enflo, who was presented with the live goose in a ceremony that was broadcast throughout Poland.
The café building now houses the Universal Bank at the street address of 27 Taras Shevchenko Prospekt.*Wik
1962 The first potential US women in space, often called the Mercury 13 in comparison to the original Mercury 7 astronauts would get a hearing in congress beginning on this day. The house convened public hearings before a special Subcommittee on Science and Astronautics. Significantly, the hearings investigated the possibility of gender discrimination a two full years before the Civil Rights Act of 1964 made that illegal, making these hearings a marker of how ideas about women's rights permeated political discourse even before they were enshrined in law. The hearings would abruptly be terminated at lunch the next day. In less than a year, Soviet cosmonaut Valentina Tereshkova became the first woman in space on June 16, 1963. In response, Clare Boothe Luce published an article in Life criticizing NASA and American decision makers. By including photographs of all thirteen Lovelace finalists, she made the names of all thirteen women public for the first time. (The Time issue is available at Google Books here. Astronaut Sally Ride became the first American woman in space in 1983 on STS-7. *Wik

1969 New York Times Apologizes for ridicule of Robert H. Goddard. and his report, “A Method of Reaching Extreme Altitudes,” published by the Smithsonian press in 1920.
In a famously nasty 1920 editorial, The New York Times ridiculed his ideas about rocketry, declaring that his claim that a rocket could fly in the vacuum of space would “deny a fundamental law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are licensed to do that.”*discovermagazine.com
(On July 17, 1969, as Apollo 11 was racing moonward, the Times published a gently self-mocking correction: “Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th Century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error.”)
1997 "You DON'T have mail"... A programming error temporarily threw the Internet into disarray in a preview of the difficulties that inevitably accompany a world dependent on e-mail, the World Wide Web, and other electronic communications.
At 2:30 a.m. Eastern Daylight Time, a computer operator in Virginia ignored alarms on the computer that updated Internet address information, leading to problems at several other computers with similar responsibilities. The corruption meant most Internet addresses could not be accessed, resulting in millions of unsent e-mail message. *This Day in History, Computer History Museum
2011 NASA's Dawn Spacecraft Enters Orbit Around Vesta
NASA's Dawn spacecraft on Saturday became the first probe ever to enter orbit around an object in the main asteroid belt between Mars and Jupiter. *NASA
1698 Pierre Louis de Maupertuis, (Saint-Malo, 17 July 1698 – Basel, 27 July 1759) developer of the principle of least action. *VFR a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great.
Maupertuis made an expedition to Lapland to determine the shape of the earth. He is often credited with having invented the principle of least action; a version is known as Maupertuis' principle – an integral equation that determines the path followed by a physical system. His work in natural history has its interesting points, since he touched on aspects of heredity and the struggle for life.*Wik (he died in the home of Johann II Bernoulli, whose death occurred on this same date, (see Deaths)
John S. Wilkins@john_s_wilkins pointed out in a tweet that "Maupertuis was the first scientific evolutionist, 7 years after first edition of Systema Naturae."
1752 Barnaba Oriani (July 17 1752 - November 12 1832) Italian geodesist, astronomer and scientist. After getting his elementary education in Carignano, he went on to study at the College of San Alessandro in Milan, under the tutelage and with the support of the Order of Barnabus, which he later joined. After completing his studies in the humanities, physical and mathematical sciences, philosophy, and theology, he was ordained as a priest in 1775.
Oriani was a devoted friend of the Theatine monk, Giuseppe Piazzi, the discoverer of Ceres. Oriani and Piazzi worked together for thirty-seven years, cooperating on many astronomical observations.For his work in astronomy, Oriani was honored by naming asteroid (4540) "Oriani". This asteroid had been discovered at the Osservatorio San Vittore in Bologna, Italy on November 6, 1988. *TIA
1831 Victor Mayer Amédée Mannheim (17 July 1831 – 11 December 1906) was the inventor of the modern slide rule. Around 1850, he introduced a new scale system that used a runner to perform calculations. This type of slide rule became known under the name of its inventor: the Mannheim.*Wik
1837 Wilhelm Lexis (July 17, 1837, Eschweiler – October 25, 1914, Göttingen), studied data presented as a series over time thus initiating the study of time series.*SAU Although the author of an Allgemeine Volkswirtschaftslehre (general economics book) (1910) and certainly a distinguished economist, even a pioneer of Law and Economics thinking and of the study of consumption and crises, Lexis is today primarily known as a statistician, partially due to his creation of the Lexis ratio (Largely replaced by the Chi-Square test). His reputation as a demographer is underlined by the ubiquity of Lexis Diagrams, which are named for him, although primary credit for their invention belongs to Gustav Zeuner and O. Brasche (a notable example of Stigler's law of eponymy). He is also one of the founding fathers of the interdisciplinary, professional study of insurance. A Kathedersozialist, he was closely affiliated with academic policy makers in Prussia and one of Friedrich Althoff’s experts and the editor of important works on German higher education, most famously the six-volume Das Unterrichtswesen im Deutschen Reich, compiled for the St. Louis World's Fair of that year and still the key reference work for that time. *Wik
1863 Herbert Richmond (17 July 1863 in Tottenham, Middlesex, England - 22 April 1948 in Cambridge, England) studied at Cambridge and spent his whole career there, His main interest was in Algebraic Geometry. He became an honorary member of the EMS in 1930.*SAU
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| Cremona-Richmond configuration |
1868 Peter Comrie (17 July 1868 in Muthill, near Crieff, Perthshire, Scotland
Died: 20 Dec 1944 in Edinburgh, Scotland) graduated from St Andrews and after a series of teaching posts became Rector of Leith Academy. He was much involved in the EMS, becoming Secretary in 1911 and President in 1916 and 1917.*SAU
1894 Georges Henri Joseph Édouard Lemaître (17 July 1894 – 20 June 1966) was a Belgian astronomer and cosmologist, born in Charleroi, Belgium. He was also a civil engineer, army officer, and ordained priest. He did research on cosmic rays and the three-body problem. Lemaître formulated (1927) the modern big-bang theory. He reasoned that if the universe was expanding now, then the further you go in the past, the universe’s contents must have been closer together. He envisioned that at some point in the distant past, all the matter in the universe was in an exceedingly dense state, crushed into a single object he called the "primeval super-atom" which exploded, with all its constituent parts rushing away. This theory was later developed by Gamow and others.*TIS The term "big bang" was created shortly after 6:30 am GMT on BBC's The Third Program, Fred Hoyle used the term in describing theories that contrasted with his own "continuous creation" model for the Universe. "...based on a theory that all the matter in the universe was created in one big bang ... ". *Mario Livio, Brilliant Blunders
He was the first to theorize that the recession of nearby galaxies can be explained by an expanding universe, which was observationally confirmed soon afterwards by Edwin Hubble.He first derived "Hubble's law", now called the Hubble–Lemaître law by the IAU, and published the first estimation of the Hubble constant in 1927, two years before Hubble's article. Lemaître also proposed the "Big Bang theory" of the origin of the universe, calling it the "hypothesis of the primeval atom", and later calling it "the beginning of the world".*Wik
Cosmic Anniversary: 'Big Bang Echo' Discovered 50 Years Ago ...
On May 20, 1964, American radio astronomers Robert Wilson and Arno Penzias discovered the cosmic microwave background radiation (CMB), the ancient light that began saturating the universe 380,000 years after its creation.
1909 Geoffrey Walker (17 July 1909 - 31 March 2001) studied at Oxford and Edinburgh. He taught at Imperial College London, Liverpool and Sheffield before returning to Liverpool as Professor of Pure Mathematics. He worked on Differential Geometry, Relativity and Cosmology.*SAU Walker was an accomplished geometer, but he is best remembered today for two important contributions to general relativity. Together with H. P. Robertson, the well known Robertson-Walker metric for the Friedmann-Lemaître-Robertson-Walker cosmological models, which are exact solutions of the Einstein field equation. Together with Enrico Fermi, he introduced the notion of Fermi-Walker differentiation.*Wik
1920 Gordon Gould (July 17, 1920 – September 16, 2005) American physicist who coined the word "laser" from the initial letters of "Light Amplification by Stimulated Emission of Radiation." Gould was inspired from his youth to be an inventor, wishing to emulate Marconi, Bell, and Edison. He contributed to the WWII Manhattan Project, working on the separation of uranium isotopes. On 9 Nov 1957, during a sleepless Saturday night, he had the inventor's inspiration and began to write down the principles of what he called a laser in his notebook Although Charles Townes and Arthur Schawlow, also successfully developed the laser, eventually Gould gained his long-denied patent rights. *TIS
1944 Krystyna M. Kuperberg (born Krystyna M. Trybulec; 17 July 1944) is a Polish-American mathematician who currently works as a professor of mathematics at Auburn University, where she was formerly an Alumni Professor of Mathematics.
by P. A. Schweitzer, and on an example of a dynamical system on R^3 constructed previously by the authors].The solution to that problem led to her 1993 work in which she constructed a smooth counterexample to the Seifert conjecture. She has since continued to work in dynamical systems. *Wik
1790 Johann II Bernoulli died (28 May 1710 in Basel, Switzerland - 17 July 1790 in Basel, Switzerland) Johann II Bernoulli (also known as Jean), the youngest of the three sons of Johann Bernoulli. He studied law and mathematics, and, after travelling in France, was for five years professor of eloquence in the university of his native city. In 1736 awarded the prize of the French Academy for his suggestive studies of Aether On the death of his father he succeeded him as professor of mathematics. He was thrice a successful competitor for the prizes of the Academy of Sciences of Paris. His prize subjects were, the capstan, the propagation of light, and the magnet. He enjoyed the friendship of P. L. M. de Maupertuis, who died under his roof (July 27, 1759) while on his way to Berlin. He himself died in 1790. His two sons, Johann and Jakob, are the last noted mathematicians of the Bernoulli family. *Wik
1899 Charles Graves (6 December 1812 – 17 July 1899) was an Irish mathematician, academic, and clergyman. He was president of the Royal Irish Academy (1861–1866). He served as dean of the Chapel Royal at Dublin Castle, and later as Bishop of Limerick, Ardfert and Aghadoe. He was the brother of both the jurist and mathematician John Graves, and the writer and clergyman Robert Perceval Graves. He and his brother John are credited with inspiring Hamiton to invent the quaternions.
1912 Jules Henri Poincare (29 April 1854 – 17 July 1912) died very suddenly from an embolism while dressing, in his 59th year. *VFR French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.
The Poincaré group used in physics and mathematics was named after him.*Wik
His Poincaré Conjecture holds that if any loop in a given three-dimensional space can be shrunk to a point, the space is equivalent to a sphere. Its proof remains an unsolved problem in topology.*TIS
His family gravestone at Cimetière de Montparnasse in Paris is covered with coins, flowers, and notes. One telling him that "It has been proven."
1917 Giuseppe Veronese (7 May 1854 – 17 July 1917) Although his work was severely criticised as unsound by Peano, he is now recognised as having priority on many ideas that have since become parts of transfinite numbers and model theory, and as one of the respected authorities of the time, his work served to focus Peano and others on the need for greater rigor.
He is particularly noted for his hypothesis of relative continuity which was the foundation for his development of the first non-archimedean linear continuum.*SAU
1944 William James Sidis (April 1, 1898 – July 17, 1944) an American child prodigy with exceptional mathematical and linguistic abilities. He became famous first for his precocity, and later for his eccentricity and withdrawal from the public eye. He avoided mathematics entirely in later life, writing on other subjects under a number of pseudonyms. The difficulties Sidis encountered in dealing with the social structure of a collegiate setting may have shaped opinion against allowing such children to rapidly advance through higher education in his day.*Wik
1963 Bevan Braithwaite Baker (1890 in Edinburgh, Scotland - 1 July 1963 in Edinburgh, Scotland) graduated from University College London. After service in World War I he became a lecturer at Edinburgh University and was Secretary of the EMS from 1921 to 1923. He left to become Professor at Royal Holloway College London.*SAU
1979 Roland George Dwight Richardson (born May 14, 1878, Dartmouth, Nova Scotia; died July 17, 1949, Antigonish, Nova Scotia) was a prominent Canadian-American mathematician chiefly known for his work building the math department at Brown University and as Secretary of the American Mathematical Society.
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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