Sunday, 9 August 2026

A Bird in the Hand is worth... a Pigeon Hole Principle

 



Sometimes problems that seem very hard, can be very easy if they are viewed in the right way, and one of those easy ways to make some hard problems manageable is the Pigeon-Hole Principle. Over the last few weeks seems like lots of problems involving this idea have shown up, so I thought I would bring it to you.
The basic idea is so easy any sixth grader would agree; if you have two boxes, and you are going to put three balls in the boxes, then at least one box will get more than one ball..... "well, Duh!" they answer... and yet... it seems easier to apply than it might be. Now that you know the secret, try these two problems. I'll post the answer down lower on the page where you must not look until you take a few minutes to ponder the problems.
Here is the first from a recent blog I read: "39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats."
This one seems tougher, but really isn't, it just requires a different way of thinking. "Suppose you pick six unique integers from 1 to 1000. Prove that at least two of them must have a difference that is a multiple of five.
Before I give you the answers, I will throw in a little cultural information that may amuse and entertain you. The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. In a discussion on a history group a few years ago Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portuguese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet" that is 'Zasada szufladkowa Dirichleta' ". You just can't have TOO many names for a really useful idea.
Ok, The Proofs... for number one... Suppose you handed each person a number that was how many seats they needed to move to the right to find their assigned seat. Since no one is at the right seat, the number can not be zero or thirty-nine. SO each of the people has a number between 1 and 38...wait, there are 39 people...two of them (at least) must be the same distance away from their assigned seats.... admit it…..that’s pretty cool. (I have a slight question about whether this actually proves the solution of "rotating the table" to put two in their correct seats.  Suppose we know that persons A and B are in each others seats and 5 seats apart.  Rotating the table five seats in either direction would only put one of them in their correct seat.  Maybe all we proved is that there are , at least, two people who are the same distance from their seat.  If we had specified how far away in a clockwise direction they are from their correct seat, we would have a solution to the rotation of the table problem.)
For number two it is sort of the same idea, but you have to think about how much each number would have for a remainder if you divided them by five. The only possible choices are 0, 1, 2, 3, or 4... , five different remainders, but there are six numbers, so two of them have the same remainder...and two numbers that have the same remainder on division by five, are a multiple of five apart.... think of 1,6, 11, etc for remainders of one. If you want to read more about how remainders can play a part in solving problems, see my blog on "casting out sevens" (And other primes) 

And for some history about this beautiful problem solving idea, see this.

On This Day in Math - August 9

 


You don't understand anything until you learn it more than one way.

— Marvin Minsky


The 221st day of the year; 221 the sum of consecutive prime numbers in two different ways 221 = (37 + 41 + 43 + 47 + 53) = (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41)

And of course, 221 is the product of consecutive primes, 13 x 17
Because 13 and 17 are both 4n+1 primes, they are the sum of two squares (13= 3^2 + 2^2 and 17 = 4^2 + 1^2), these can be used to construct two ways that 221 can be shown as the sum of two primes. (221 = (3*4+1x2)^2 + (4x2-1x3)^2 = 14^2 + 5^2 and by changing +/-  (3x4-1x2)^2 + (4x2+1x3)^2 = 10^2 + 11^2 = 221. Known around 1st century AD by Diophontas.

221221 + 122 is prime, it is the only known number greater than one with this property.

221 is the number of 7-vertex Hamiltonian planar graphs ( a graph that allows a closed path that visits each node exactly once.)


221 is an arithmetic number, the mean of its divisors is a whole number; (1 + 13 + 17 + 221) /4 = 63

221 in hexdecimal (base 16) is given  by (DD) or (13, 13) 13*16 + 13.

221 is also a palindrome in base 7 (434) and base 11 (191).


See More Math Facts for every Year Day here.




EVENTS

0975 "The Sun was eclipsed . . . . Some people say that it was entirely total. During the hours mao and ch'en (some time between 5 and 9 h) it was all gone. It was the color of ink and without light. All the birds flew about in confusion and the various stars were all visible. There was a general amnesty (on account of the eclipse)." From: Nihon Kiryaku. "At the hour ch'en (7-9 h), the Sun was eclipsed; it was completely total. All under heaven became entirely dark and the stars were all visible."
From: Fuso Ryakki. "The Sun was eclipsed; it was all gone. It was like ink and without light. The stars were all visible (or: stars were visible in the daytime)." From: Hyaku Rensho. These three Japanese quotations refer to a total solar eclipse of 9 August AD 975. Quoted in Historical Eclipses and Earth's Rotation, by F Richard Stephenson, Cambridge University Press, 1997, pages 267 and 268. *NSEC

Night Sky Map view of totality over the US city of Dallas—during the upcoming solar eclipse of April 8, 2024—the Sun is shown to be in front of the constellation Pisces. It forms a nearly straight line with the bright planets Venus (right) and Jupiter (left) in the sky, revealing the path of the ecliptic.





1207 An educational institution is founded for the study of the works of Bhaskara II, an Indian mathematician and astronomer.*VFR
Bhaskara II was rightly achieved an outstanding reputation for his remarkable contribution. In 1207 an educational institution was set up to study Bhaskaracharya's ("Bhaskara the teacher") works. A medieval inscription in an Indian temple reads: Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with fame and religious merit, he is like the crest on a peacock. It is from this quotation that the title of Joseph's book comes. *SAU



1654 Fermat to Carcavi "Monsieur, I was overjoyed to have had the same thoughts as those of M. Pascal, for I greatly admire his genius and I believe him to be capable of solving any problem he attempts. The friendship he offers is so dear to me and so precious that I shall not scruple to take advantage of it in publishing an edition of my Treatises. If it does not shock you, you could both help in bringing out this edition, and I suggest that you should be the editors: you could clarify or augment what seems too brief and thus relieve me of a care which my work prevents me from taking. I would like this volume to appear without my name even, leaving to you the choice of designation which would indicate the author, whom you could qualify simply as a friend." *York Univ Hist of Stats.

Fermat



Carcavi



1658 Simon Douw obtains a patent for a pendulum clock that will draw Huygen’s attack in Horologium. “Today, no clock by Simon Douw is known; I find that most curious, it is as if he has been excised from history, deliberately. Dutch Court papers described Douw as "City clockmaker of Rotterdam... a master in the art of great tower, domestic or office clocks", ("en meester in de kunst van groote Toorn, Camer ofte Comptoirwerken"). Yet his mechanical insights. his escapement, also his drive mechanisms, are best, and now only, revealed by his Patent Grant on August 9th, 1658, and by the evidence and judgement in a claim and counterclaim started in the Provinces of Holland and West Friesland, but then referred to the Court of The Netherlands in October 1658, with a Judgement by Consent on December 5th, 1658. And that case went entirely in Douw's favor, against the highly favored joint Complainants Huygens and Coster.
In itself, that is remarkable. Huygens, the Noble patrician, the most famous Dutch scientist, and the self-professed inventor of the pendulum clock, who had in the course of this trial published "Horologium"(where he also predicted an easy victory over Douw in the Courts), was forced by the judges to settle the case rather than face unfavorable verdict; also to concede Consent; also one-third Royalties to Douw. It would have been a crushing humiliation for Huygens, the seed of his libels. Subsequently, the Lower Court of Holland, Zeeland and Friesland confirmed to Douw, on December 16th and 19th 1658, their Upper Court's judgement by consent”. * From Keith Piggot

Huygen's first pendulum clock:





1663  Christiaan Huygens ‘saw performed at Gresham College a notable experiment upon a dog, to which an opening having been made in the body, the whole spleen was cut out, having first joined up the large veins. After the hole was sewn up it went about as if nothing bad had happened; and it is found that they survive very well after despite being deprived of that part.’ *Hugh Aldersey-Williams@HoooAW

The experiment was most likely performed by Robert Hooke.  While Hooke’s own diary doesn't begin until 1672, we know from other Royal Society records (and from accounts like Huygens’s) that he performed many such experiments in the early 1660s.  

Hooke conducted a similar vivisection experiment, involving removal or impairment of lung function in a dog.

According to historical records, Hooke did an experiment where a dog’s chest was opened, its lungs slit, and then ventilated continuously using bellows. Despite the lung impairment, the dog remained alive and responsive until the artificial ventilation stopped. In fact, the operation even included cutting off part of the lungs, yet circulation continued as long as fresh air was supplied 




1895 Percival Lowell, convinced about intelligent beings on Mars, published an article in The Atlantic Monthly about how lucky they are to have low gravity. "LUCK OF THE BEING WHO LIVES ON MARS; He Can Do More Work Much More Easily than Man on Earth." *HT Paul Halpern‏ @phalpern




1898 Rudolf Diesel patents an internal combustion engine in the US (filed July 15, 1895.) "My invention has reference to improvements in apparatus for regulating the fuel supply in slow-combustion motors" *Google.com


1975 To display Mexican-Lebanese friendship, Mexico issued a stamp of the Teacher’s Monument in Mexico City by I. Naffa al Rozzi, which shows Cadmus, a mythical Phoenician, teaching the alphabet.

Named Al Maestro, meaning The Master, the monument is located at the Papalote Children’s Museum (Papalote Museo del Nino) in Mexico City, and depicts Cadmus and Europa teaching the alphabet to Greek students (and to the world).




1991  First E-mail Sent from Space  Using a Mac Portable aboard the Space Shuttle Atlantis, the first e-mail from space is sent to Earth. Two astronauts on the spacecraft, James Adamson and Shannon Lucid, wrote, “Hello Earth! Greetings from the STS-43 Crew. This is the first AppleLink from space. Having a GREAT time, wish you were here,...send cryo and RCS! Hasta la vista, baby,...we'll be back!” The message was transmitted to the Johnson Space Center in Houston, Texas.  *CHM




BIRTHS


1537 Franciscus Barocius (9 August 1537 – 23 November 1604) born. In 1560 he published the first important translation of Proclus’ commentary on the first book of Euclid’s Elements. In 1587 he was brought before the Inquisition on charges of sorcery, more particularly of having caused a torrential rainstorm in Crete. *VFR

Barozzi translated many works of the ancients, in addition to Proclus’s edition of Euclid's Elements (published in Venice in 1560), as well as mathematical works by Hero, Pappus of Alexandria, and Archimedes.

He also wrote Rythmomachia (1572), a work that is based on the mathematical game of the same name, also known as "The Philosophers' Game."

Rithmomachia is an early European mathematical board game. Its earliest known description dates from the eleventh century. The name comes loosely from Greek and means "The Battle of the Numbers."[a] The game is somewhat like chess except that most methods of capture depend on the numbers inscribed on each piece.

The game was used as an educational tool that teachers could introduce while teaching arithmetic as part of the quadrivium to those in Western Europe who received a classical education during the medieval period. *Wik





1602 Gilles de Roberval (August 9, 1602 – October 27, 1675)(His date of birth is given as 8th, 9th and 10th in various sources) was a French scientist who developed powerful methods in the early study of integration.*SAU Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, which would today be solved by calculus. He worked on the quadrature of surfaces and the cubature of solids, which he accomplished, in some of the simpler cases, by an original method which he called the "Method of Indivisibles"; but he lost much of the credit of the discovery as he kept his method for his own use, while Bonaventura Cavalieri published a similar method which he independently invented.
Another of Roberval’s discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions. (The limacon was named by Roberval in 1650 when he used it as an example of his methods of drawing tangents.)

He also discovered a method of deriving one curve from another, by means of which finite areas can be obtained equal to the areas between certain curves and their asymptotes. To these curves, which were also applied to effect some quadratures, Evangelista Torricelli gave the name "Robervallian lines."
(He also wrote a) work on the system of the universe, in which he supports the Copernican heliocentric system and attributes a mutual attraction to all particles of matter. *Wik
 I was recently informed (2018) by Vincent Panteloni that in France, the balance scale, "We refer to such a scale by saying 'une balance de Roberval'".


1757 Thomas Telford (9 August 1757 Glendinning, Westerkirk, Eskdale, Dumfriesshire, Scotland - 2 September 1834 (aged 77) 24 Abingdon Street, Westminster, London) He is the founder of modern bridge construction, his crowning achievement being the Menai suspension bridge in Wales. Do you know the shape of the cables on a suspension bridge? *VFR


Menai Suspension Bridge *Wik


Telford's reputation in Shropshire led to his appointment in 1793 to manage the detailed design and construction of the Ellesmere Canal, linking the ironworks and collieries of Wrexham via the north-west Shropshire town of Ellesmere, with Chester, utilising the existing Chester Canal, and then the River Mersey.

Among other structures, this involved the spectacular Pontcysyllte Aqueduct over the River Dee in the Vale of Llangollen, where Telford used a new method of construction consisting of troughs made from cast iron plates and fixed in masonry. Extending for over 1,000 feet (300 metres) with an altitude of 126 ft (38 m) above the valley floor, the Pontcysyllte Aqueduct consists of nineteen arches, each with a 45 ft (14 m) span. Being a pioneer in the use of cast-iron for large scaled structures, Telford had to invent new techniques, such as using boiling sugar and lead as a sealant on the iron connections. Eminent canal engineer William Jessop oversaw the project, but he left the detailed execution of the project in Telford's hands. The aqueduct was designated a UNESCO World Heritage Site in 2009. *Wik

A canal boat traverses the Pontcysyllte aqueduct in North Wales



1776 Count Amedeo Avogadro (9 August 1776, Turin, Piedmont – 9 July 1856) Italian chemist and physicist who found that at the same temperature and pressure equal volumes of all perfect gases contain the same number of particles, known as Avogadro's Law (1811) leading to the Avogadro's constant being 6.022 x 1023 units per mole of a substance. He realized the particules could be either atoms, or more often, combinations of atoms, for which he coined the word "molecule." This explained Gay-Lussac's law of combining volumes (1809). Further, Avogadro determined from the electrolysis of water that it contained molecules formed from two hydrogen atoms for each atom of oxygen, by which the individual oxygen atom was 16 times heavier than one hydrogen atom (not 8 times as suggested earlier by Dalton.) The Italian, Romano Amadeo Carlo Avogadro, had suggested [in 1811] that all gases have the same number of molecules in a given volume. Loschmidt figured out [in 1865] how many molecules that would be. John D. Cook suggested that maybe it should be called Loschmidt's constant, and pointed out three interesting coincidences involving Avogadro's Constant:
NA is approximately 24! (i.e., 24 factorial.)
The mass of the earth is approximately 10 NA kilograms.
The number of stars in the observable universe is 0.5 NA.
*John D. Cook, The Endeavour Blog




1819 Jonathan Homer Lane (August 9, 1819, Geneseo, New York – May 3, 1880, Washington D.C.) U.S. astrophysicist who was the first to investigate mathematically the Sun as a gaseous body. His work demonstrated the interrelationships of pressure, temperature, and density inside the Sun and was fundamental to the emergence of modern theories of stellar evolution.*TIS




1861 Dorothea Klumpke Roberts (August 9, 1861 in San Francisco – October 5, 1942 in San Francisco) was an American astronomer. She was the Director of the Bureau of Measurements at the Paris Observatory and was made a Chevalier de la Légion d'Honneur, or a Knight of the National Order of the Legion of Honor.

In 1877, Klumpke moved to Paris, France, while her four sisters attended schools in Germany and Switzerland. She studied at the University of Paris. She began by studying music, but later turned to astronomy. She earned her bachelor's degree in 1886 and her PhD in 1893, with her dissertation focusing on the rings of Saturn. In 1887, she began working at the Paris Observatory alongside Guillaume Bigourdan and Lipót Schulhof, and later astrophotographers Paul and Prosper Henry. Her work consisted of measuring star positions, processing astrophotographs, and studying stellar spectra and meteorites.

In 1886, Sir David Gill proposed an atlas of the heavens. The idea received enthusiastic support, especially from the Director of the Paris Observatory, Admiral Amédée Mouchez, who suggested an international meeting in Paris. This led to the Carte du Ciel project, which required photographing the entire sky and showing stars as faint as the 14th magnitude. The Paris Observatory was to do a major portion of the sky as its contribution. It was also envisioned that a catalogue of all the stars to the 11th magnitude be drawn up.

Klumpke was appointed the Director of the Bureau of Measurements (Bureau des Mesures) at the Paris Observatory, a position she held for a decade. She supervised several other women scientists during this time.

In 1896, she sailed to Norway on the Norwegian vessel Norse King, to observe the solar eclipse of August 9, 1896. There, she became acquainted with Dr. Isaac Roberts, a 67-year-old Welsh widower, entrepreneur, and astronomer, who had become a pioneer in astrophotography. He had also attended the Paris Carte du Ciel Congress.

In 1899, astronomers had predicted a great meteor shower now known as the Leonids. The French chose Klumpke to be the one to ride in a balloon to observe the shower. The shower turned out to be a complete failure.

In 1901, Dorothea Klumpke and Isaac Roberts were married and moved to his home in Sussex, England. Roberts left her job at the Paris Observatory to be with her husband, whom she assisted in a project to photograph all 52 of the Herschel "areas of nebulosity." Their marriage lasted until Isaac's death in 1904. Roberts inherited all his astronomical effects and a considerable fortune. *Wik




1911 William Alfred "Willie" Fowler (August 9, 1911 – March 14, 1995)  American astrophysicist. He should not be confused with the British astronomer Alfred Fowler.

Fowler won the Henry Norris Russell Lectureship of the American Astronomical Society in 1963, the Eddington Medal in 1978, the Bruce Medal in 1979, and the Nobel Prize for Physics in 1983 for his theoretical and experimental studies of the nuclear reactions of importance in the formation of the chemical elements in the universe (shared with Subrahmanyan Chandrasekhar). *TIA




1908 Mary Golda Ross (August 9, 1908 – April 29, 2008) was the first known Native American female engineer, and the first female engineer in the history of Lockheed. She was one of the 40 founding engineers of the renowned and highly secretive Skunk Works project at Lockheed Corporation. She worked at Lockheed from 1942 until her retirement in 1973, where she was best remembered for her work on aerospace design – including the Agena Rocket program – as well as numerous "design concepts for interplanetary space travel, crewed and uncrewed Earth-orbiting flights, the earliest studies of orbiting satellites for both defense and civilian purposes." In 2018, she was chosen to be depicted on the 2019 Native American $1 Coin by the U.S. Mint celebrating American Indians in the space program. *Wik


 
1919 Leona Woods (August 9, 1919 – November 10, 1986), later known as Leona Woods Marshall and Leona Woods Marshall Libby, was an American physicist who helped build the first nuclear reactor and the first atomic bomb.
At age 23, she was the youngest and only female member of the team which built and experimented with the world's first nuclear reactor (then called a pile ), Chicago Pile-1, in a project led by her mentor Enrico Fermi. In particular, Woods was instrumental in the construction and then utilization of geiger counters for analysis during experimentation. She was the only woman present when the reactor went critical. She worked with Fermi on the Manhattan Project, and, together with her first husband John Marshall, she subsequently helped solve the problem of xenon poisoning at the Hanford plutonium production site, and supervised the construction and operation of Hanford's plutonium production reactors.
After the war, she became a fellow at Fermi's Institute for Nuclear Studies. She later worked at the Institute for Advanced Studies in Princeton, New Jersey, the Brookhaven National Laboratory, and New York University, where she became a professor in 1962. Her research involved high-energy physics, astrophysics and cosmology. In 1966 she divorced Marshall and married Nobel laureate Willard Libby. She became a professor at the University of Colorado, and a staff member at RAND Corporation. In later life she became interested in ecological and environmental issues, and she devised a method of using the isotope ratios in tree rings to study climate change. She was a strong advocate of food irradiation as a means of killing harmful bacteria. *Wik

1927 Marvin Minsky (August 9, 1927 - January 24, 2016 (aged 88)) Biochemist and the founder of the MIT Artificial Intelligence Project. Marvin Minsky has made many contributions to AI, cognitive psychology, mathematics, computational linguistics, robotics, and optics. He holds several patents, including those for the first neural-network simulator (SNARC, 1951), the first head-mounted graphical display, the first confocal scanning microscope, and the LOGO "turtle" device. His other inventions include mechanical hands and the "Muse" synthesizer for musical variations (with E. Fredkin). In recent years he has worked chiefly on imparting to machines the human capacity for commonsense reasoning. *TIS He died in Boston of a cerebral hemorrhage .

Logo Turtle





1939  Alan Baker FRS (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory.
His interests were in number theory, transcendence, linear forms in logarithms, effective methods, Diophantine geometry and Diophantine analysis.
Baker generalised the Gelfond–Schneider theorem, which itself is a solution to Hilbert's seventh problem.
In 2012 he became a fellow of the American Mathematical Society. He has also been made a foreign fellow of the National Academy of Sciences, India.




1940 Linda Goldway Keen (8 August 1940- ) In addition to studying Riemann surfaces, Keen has worked in hyperbolic geometry, Kleinian groups and Fuchsian groups, complex analysis, and hyperbolic dynamics. In the field of hyperbolic geometry, she is known for the Collar lemma.
Keen has worked at the Institute for Advanced Study, Hunter College, University of California at Berkeley, Columbia University, Boston University, Princeton University, and the Massachusetts Institute of Technology, as well as at various mathematical institutes in Europe and South America. After her initial appointment in 1965, in 1974 Keen was promoted to Full Professor at Lehman College and the CUNY Graduate Center.
Keen served as president of the Association for Women in Mathematics during 1985-1986 and as vice-president of the American Mathematical Society during 1992-1995. She served on the Board of Trustees of the American Mathematical Society from 1999-2009 and as Associate Treasurer from 2009-2011. In 1975, she presented an AMS invited address and in 1989 she presented an MAA joint invited address. In 1993 she was selected as a Noether Lecturer. *Wik




1943 Jacques Lewiner, (9 August, 1943 - ) is a French physicist and inventor. He is Professor and Honorary Scientific Director of École supérieure de physique et de chimie industrielles de la ville de Paris (ESPCI ParisTech).
His works have been devoted to electrical insulators and particularly electrets, instrumentation and sensors, for instance in medical imaging, or on the improvement of telecommunication networks.
He has filed a large number of patent applications leading to industrial development, either through licenses granted to industrial companies or through start-up companies often created with former students or researchers. He has participated in the creation of various technology oriented start up companies, for instance Inventel, specializing in Telecommunications, Finsécur which develops and markets fire detection systems, Sculpteo which is an online 3D printing platform, Roowin in the field of chemical synthesis and Cynove in embedded electronics devices. Most of these companies have experienced a strong growth. For instance Inventel, which was the French leader for multimedia gateways was bought by Thomson SA in 2005.
Lewiner is laureate of the French Academy of Sciences in 1990, Knight in the National Order of the Legion of Honor, member of the French Academy of Technologies since 2005 and Honorary Fellow of the Technion. *Wik





DEATHS




1853 Josef-Maria Hoëné de Wronski (23 August 1776 - 9 August 1853)wrote on the philosophy of mathematics. *SAU He wrote exclusively in French, desirous that his ideas, of whose immortality he was convinced, should be accessible to all; he worked, he said, "through France for Poland." He published over a hundred works, and left many more in manuscript. When dying in the seventy-fifth year of his life, he exclaimed: "God Almighty, there's still so much more I wanted to say!"
In science, Hoene-Wroński set himself maximal tasks: the complete reform of philosophy and of mathematics, astronomy, technology. He not only elaborated a system of philosophy, but applications to politics, history, economics, law, psychology, music, pedagogy. It was his aspiration to reform human knowledge in an "absolute, that is, ultimate" manner.
Though during his lifetime nearly all his work was dismissed as nonsense, some of it has come in later years to be seen in a more favorable light. Although nearly all his grandiose claims were in fact unfounded, his mathematical work contains flashes of deep insight and many important intermediary results. Most significant was his work on series. He had strongly criticized Lagrange's use of infinite series, introducing instead a novel series expansion for a function. His criticisms of Lagrange were for the most part unfounded, but the coefficients in Wroński's new series were found to be important after his death, forming the determinants now known as the Wronskians (the name was given them by Thomas Muir in 1882).
The level of Wroński's scientific and scholarly accomplishments, and the amplitude of his objectives, placed Wroński in the first rank of European metaphysicians in the early 19th century. But the abstractness, formalism and obscurity of his thought, the difficulty of his language, his boundless self-assurance, his uncompromising judgments of others—alienated. He was perhaps the most original of the Polish metaphysicians, but others were more representative of the Polish outlook. *Wik



1929 Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.
In 1917–1920 Fatou created the area of mathematics which is called holomorphic dynamics (Fatou 1919, 1920, 1920b). It deals with a global study of iteration of analytic functions. He was the first to introduce and study the set which is called now the Julia set. (The complement of this set is sometimes called the Fatou set). Some of the basic results of holomorphic dynamics were also independently obtained by Gaston Julia and Samuel Lattes in 1918. Holomorphic dynamics has experienced a strong revival since 1982 because of the new discoveries of Dennis Sullivan, Adrien Douady, John Hubbard and others. In 1926, Fatou pioneered the study of dynamics of transcendental entire functions (Fatou 1926), a subject which is intensively developing at this time.*Wik

Julia sets for 𝑧^2+0.7885 𝑒^(𝑖𝑎) ,where a ranges from 0 to 2𝜋

*Wik


1932 John Charles Fields died (May 14, 1863 - August 9, 1932). In his will he left funds for an international medal for contributions to mathematics. The International Congress of Mathematicians in Zurich in 1932 adopted the proposal, and the first Fields Medals were awarded at the Oslo Congress in 1936 to Lars Ahlfors, age 29 of Harvard, and Jesse Douglas, age 39 of Massachusetts Institute of Technology. 
 
It became the most prestigious award for mathematicians, often referred to as the equivalent of a Nobel Prize for mathematicians. As a professor at the University of Toronto, he had worked to bring the International Congress of Mathematicians to Toronto (1924). The Congress was so successful that afterward there was a surplus of about\( $2,500\) which Fields, as chairman of the organizing committee, proposed be used to fund two medals to be awarded at each of future Congresses. This was approved on 24 Feb 1931. He died the following year, leaving \($47,000\) as additional funding for the medals, which have been awarded since 1936.*TIS


1969 Cecil Frank Powell (5 December 1903 – 9 August 1969) British physicist and winner of the Nobel Prize for Physics in 1950 for his development of the photographic method of studying nuclear processes and for the resulting discovery of the pion (pi-meson), a heavy subatomic particle. The pion proved to be the hypothetical particle proposed in 1935 by Yukawa Hideki of Japan in his theory.*TIS




1994 Helena Rasiowa (June 20, 1917 – August 9, 1994) worked in algebraic logic and the mathematical foundations of computer science.*SAU
Rasiowa became strongly influenced by Polish logicians. She wrote her Master's thesis under the supervision of Jan Łukasiewicz and Bolesław Sobociński. In 1944, the Warsaw Uprising broke out and consequently Warsaw was almost completely destroyed. This was not only due to the immediate fighting, but also because of the systematic destruction which followed the uprising after it had been suppressed. Rasiowa's thesis burned with the whole house. She herself survived with her mother in a cellar covered by the ruins of the demolished building.
After the war, Polish mathematics began to recover its institutions, its moods, and its people. Those who remained considered their duty to be the reconstruction of Polish universities and the scientific community. One of the important conditions for this reconstruction was to gather all those who could participate in re-creating mathematics. In the meantime, Rasiowa had accepted a teaching position in a secondary school. That is where she met Andrzej Mostowski and came back to the university. She re-wrote her Master's thesis in 1945 and in the next year she started her academic career as an assistant at the University of Warsaw, the institution she remained linked with for the rest of her life.

At the university, she prepared and defended her PhD thesis, Algebraic Treatment of the Functional Calculi of Lewis and Heyting, in 1950 under the guidance of Prof. Andrzej Mostowski. This thesis on algebraic logic initiated her career contributing to the Lwów–Warsaw school of logic: In 1956, she took her second academic degree, doktor nauk (equivalent to habilitation today) in the Institute of Mathematics of the Polish Academy of Sciences, where between 1954 and 1957, she held a post of associate professor, becoming a professor in 1957 and subsequently Full Professor in 1967. For the degree, she submitted two papers, Algebraic Models of Axiomatic Theories and Constructive Theories, which together formed a thesis named Algebraic Models of Elementary Theories and their Applications.

Rasiowa was pivotal in her role for inviting many international mathematicians, especially logicians, to Poland in the late '70s and early '80s, despite the country's instability at the time. According to Japanese mathematician Hiroakira Ono, she enabled collaboration through her iron will.*Wik



2006 James Alfred Van Allen (September 7, 1914 – August 9, 2006) American physicist who discovered the Earth's magnetosphere, two toroidal zones of radiation due to trapped charged particles encircling the Earth (also known as the Van Allen radiation belts). During WWII he gained experience miniaturizing electronics, such as in the proximity fuse of a missile. After the war, he studied cosmic radiation, taking advantage of the unused German stock of V2 rockets launched into the outer regions of the atmosphere, carrying research devices using radio to relay back the data gathered. He was also involved in the early U.S. space program, and he had radiation measuring instruments on the first U.S. satellite, Explorer 1, launched 31 Jan 1958 with follow-up carried out by satellites Explorer 3 and 4 later the same year.*TIS



2007 Graham Robert Allan (August 13, 1936 Southgate, London - August 9, 2007 (aged 70)) was an English mathematician, specializing in Banach algebras. He was a reader in functional analysis and vice-master of Churchill College at Cambridge University. 
Allan spent most of his career at Cambridge, with interludes as a Lecturer in Pure Mathematics at Newcastle University from 1967 to 1969 and as Professor of Pure Mathematics at the University of Leeds from 1970 to 1978.

Back at Cambridge, he was promoted to Reader in 1980 and was Vice-Master of Churchill College from 1990 to 1993. Allan supervised the theses of over 20 Cambridge PhD students. He retired in 2003, but continued teaching after his retirement. He died on 9 August 2007 in Cambridge.

In 1969, Allan won the Junior Berwick Prize of the London Mathematical Society.

He contributed to section III.86 in the book The Princeton Companion to Mathematics edited by Timothy Gowers, but did not live to see his article "The Spectrum" in print form published in 2008.n*Wik




2018  John David Philip Meldrum (18 July 1940 in Rabat, Morocco; died 9 August 2018 in Edinburgh, Scotland) was a British mathematician. Meldrum was an algebraist and his research was mostly related to group theory.

In 1964 he was appointed as a supernumerary fellow and college lecturer in mathematics at Emmanuel College.Meldrum received his PhD from the University of Cambridge in 1967 on the topic of "Central Series in Wreath Products". His supervisor was Derek Roy Taunt.

In 1969 he became a lecturer for mathematics at the University of Edinburgh and in 1982 he was appointed there as a senior lecturer.

He died on 9 August 2018 in Edinburgh after a battle with the Parkinson's disease. *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


Saturday, 8 August 2026

Not All Math Symbols are Equal, or How "=" Became Ubiquitous

  Another from the Archives:2008



I have a friend named Dave Refro who writes and edits questions for one of those high stakes tests that is used for admission into certain graduate programs and uses his job as an excuse for his fascination with archiving old math journal articles. Some folks garden, Dave archives. He spends hours pouring through journals and abstracts and fits together articles with a common theme. If you read almost any math discussion on line, you will probably have come across one of Dave's responses to a question with numerous links to how the question was addressed, discussed, and argued over through history.
Fortunately for me, Dave sometimes finds an article that he thinks might be of interest to me, and when he gets a stack of them, I get a big present in the mail and my wife knows I will be taking my meals in the den for a few days. In a stack of journal articles he sent recently, (THANKS Dave!) there was one particularly interesting article by Florian Cajori from 1923. In the article Cajori points out two interesting things about the equal sign that every one uses; and that is one of the interesting things he points out, is that EVERYONE uses it. Even in 1923, it was one of the most ubiquitous math symbols in the world and today there are still only about four math symbols that you could write and they would not only be understood, but written exactly the same way whether you found yourself in darkest Africa, the Far East, or downtown Los Angeles. It seems like the perfect symbol, and as Robert Recorde said when he created the symbol in 1557 in his "Whetstone of Witte", the use of "a pair of paralleles, or Gemowe(twin, from the same root as Gemini) lines of one length ... bicause noe 2 thynges can be moare equalle." In fact, Recorde's equal sign had much longer lines than is common today, sort of like == but longer.


Indeed, one wonders why it hadn't been thought of years before, and assume that it immediately became the most common of mathematical notations....ahhh, but not so. The other thing Cajori commented on that I think would surprise young students is that it took over a hundred years for the symbol to become accepted. So what symbol did mathematicians use before the good old == signs? Well, many of them used nothing. The early development of algebra occurred with a very rhetorical approach. When people wanted to write 7x+5 = 26, they would say," the product of seven and some quantity when added to five will equal twenty-six." Ok, they probably said it in Latin, and sometimes they did write numbers in place of the words for numbers, but for equals, they often wrote out the Latin aequales or some variation of it. Frequently they used abbreviations instead of full words and so "p" would stand for plus and "m" for minus...and they would shorten aequales as "aeq" or just "ae". By the time that Recorde had his inspirational stroke, lots of other people had decided THEY had a really good symbol. A pair of vertical lines, ||, was used by Xylander (Wilhelm Holzman) in his translation of Diophantus, Arithmetica only a few years later, and Regiomontanus had used a single horizontal line for equality almost a century earlier. Descartes used the symbol below in his Geometrie, which was probably drawn from the "ae" abbreviation for aequalis. and Johann Caramuel used equal lines where we would use a decimal point, so Pi would be 3==1415 etc.

 

 Descartes symbol became a popular competitor on the continent, finding favor with Huygens and the Bernoulli's, while many of the he English mathematicians, Wallis, Barrow, and Newton, followed Recorde's lead. Others used the "gemowe" lines of Recorde for other meanings, Descartes used it to mean +/- in his

So what brought the divided world into a common accord? It took a brand new idea, a revolutionary idea, the calculus. As if by divine providence, the two great minds that created the calculus, almost in unison, tended to publish their versions with a common symbol for equality, Recorde's "gemowe" lines. They disagreed on almost every other symbol they used, but in the last half of the 1600's and the early 1700's the = sign rose to world dominance. In Cajori's words, "The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption."
If you can get your students to understand how long and difficult it is to get mathematicians to accept a symbol, perhaps they will not be too surprised if their College Prof goes into a rant when they use the symbol "ln" for the natural log... and if they accept that the symbol exists (honest, they don't all accept it's use), I can't begin to imagine how they will react if you pronounce it differently than they would. My advice to students; wait for them to say it first!

On This Day in Math - August 8

    


This result is too beautiful to be false;
it is more important to have beauty in one's equations
than to have them fit experiment.

Paul A M Dirac

The 220th Day of the Year
220 is the smallest amicable number, paired with 284. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other. Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. (what is the next pair?)

If you add the sum of all the divisors of the first 16 numbers, you get 220.

220 is the the largest difference between two consecutive primes less than 100,000,000.

220 is the tenth tetrahedral number, the sum of the first ten triangular numbers, 1 + 3 + 6 + ... + 55 *Wikipedia

220 is a pseudo-perfect number, the sum of a subset of its proper divisors.  220 = 110 + 55 + 44 + 11.  Almost all abundant numbers (the sum of its proper divisors is greater than the numberare pseudoperfect, but Not quite all.  Can you find the smallest of these 'weird" abundant numbers that are not pseudoperfect. And for a real challenge, I don't think any odd number is weird...and that's odd to me.

220 is a tetrahedral number, the sum of the first ten triangular numbers.  220= 1 + 3 + 6 + ...+ 55

Finding a number whose reciprocal is equal to the sum of two other reciprocals is an important idea in many areas of math and science. 220 turns out to be the largest of a triple of that type, 1/220 + 1/180 = 1/99

Because 220/4 = 55, 220 = 56^2 - 54^2, If all the diagonals of a regular dodecagon are drawn, they divide the dodecagon into 220 regions.

220 is the sum of four consecutive primes. 47 + 53 + 59 + 61.

Interesting that if you multiply the first 220 composite numbers, it is a composite number that falls between a pair of twin primes. *Prime Curios

Every number less than 220 can be formed by the sum of divisors of 220 .

220 is the largest gap between consecutive primes less than 10^8.

And Derek Orr Points out that the aliquot(proper divisors) sequence for 220 does not end in 1. This was known to the ancients who described such numbers as amicable numbers. 220 and 228 are each the aliquote sum of the other. There are a few other numbers which have repeating sequences of three or more in a loop. The aliquot sequence of perfect numbers is fixed by their definition. There are other non-perfect numbers whose sequence eventually lands on a perfect number and then repeats that number infinitly (are until you stop calculating). and there are a few other numbers which have repeating sequences of three or more in a loop.

You can make a 5x5 magic square with consecutive numbers from 32 to 44, following the order of the standard 5x5 magic square.... or simply add 31 to each one in the primitive square.



EVENTS

1576 Laying of the cornerstone of Tycho Brahe’s observatory on the island of Hveen. *VFR

1667. John Evelyn records a visit to Secretary of Royal Society held in Tower of London; "Visited Mr. Oldenburg, a close prisoner in the Tower, being suspected of writing intelligence. I had an order from Lord Arlington, Secretary of State, which caused me to be admitted. This gentleman was secretary to our Society, and I am confident will prove an innocent person." *Diary of John Evelyn
It seems to have been some rash political comment in Oldenburg's own letters during the Dutch war in 1667 which led to his imprisonment for two months in the Tower of London. But he was soon rehabilitated, and resumed his tireless work for the Royal Society, continuing to manage its voluminous correspondence until his death in 1677.




1786 Standards for the decimal system of money established (in the USA). *VFR The first really popular English language arithmetic by an American born author was in 1788 when Nicholas Pike published A New and Complete System of Arithmetic, which he said was "Composed for the use of the citizens of the United States". Well, patriotism probably won't hurt sells in a new country. Pikes book carried endorsements from several noted persons, including the governor of Massachusetts, James Bowdoin, and Yale President Ezra Stiles. The book even included a copy of the Act of Congress of 1786 which created the U. S. Federal Money System with denominations of mills (1/1000 of a dollar), cents, dimes, dollars, and Eagles (ten dollars). With all this emphasis on the new USA, it seems strange that none of the problems in the book involved the new American money, but instead were based on the English system. [The 2nd edition, in 1797, includes in the (very long) title; "adapted to the Federal Currency by Nathaniel Lord, A.M.;Boston]"

There was never a mill coin minted, but it was a legal denomination for taxes and pricing and such, purely a bookkeeping value. 
The Coinage Act of 1792 created the half disme (spelled that way in the law) as a silver coin worth five cents.
A small batch of pattern half dismes was struck in Philadelphia, probably from Martha Washington’s donated silverware, but these were more trial issues than general circulation coins.
in 1794 regular production of the half dime began, with the Flowing Hair design, struck in silver. These remained the standard 5-cent coin until after the Civil War.

1866 – The first nickel five-cent piece (made of copper-nickel, larger in size) was issued. This was the first 5-cent coin not made of silver.







1802  We have William Herschel's Diary to thank for the exact date, unfortunately, he only paraphrased the words that passed between Napoleon and Laplace, so we have to sort out the various reports of those who were not present.  My favorite version is told by Thomas Levenson in his "The Hunt for Vulcan..." It goes like this:
Napoleon took a moment during the brief peace of 1802 to engage in a bit of intellectual banter.  He entertained a few savants - Sir William Herschel himself, the distinguished physicist Count Rumford, his minister of the interior - a chemist by profession - Jean Antoine Chaptal, and Laplace.  After engaging politenesses with Herschel, the First Consul next turned to Laplace, who had just published the third volume of Celestial Mecahanics.  Released from matters of state, Napoleon delighted in putting awkward questions to his guests, and so he told his mathematical friend that he had read Newton, and saw that his great book had mentioned God often.  But "I have perused yours, but failed to find his name even once."  Why is that, he asked? 

In the grand tradition of this story, Laplace is reported to have replied, "I have no need of that hypothesis."

the letter from Herschel



 


In 1854, metal bullet cartridges were patented by Smith & Wesson.*TIS Prior to this, cartridges were formed from paper. In 1776 a schoolmaster in Vienna named Felkel completed a factor table to 408,000 that was intended to be part of a larger work to reach several million. The tables were published at the expense of the Austrian government in the hope that subscriptions would pay for the cost. When the subscriptions failed to meet expectations, the printed volumes were supposedly used for cartridge paper. *Oystein Ore.. Number Theorey and Its History, pg 54




1876 Thomas Alva Edison, of Menlo Park, New Jersey, obtained patent #180,857 for a “method of preparing autographic stencils for printing,” the first mimeograph machine. *VFR Only those of us who have been in the classroom a loooong time will remember these handy machines (and their intoxicating aroma). (addendum; Charles Wells has advised me "The solvent in mimeo ink was castor oil.")




1900 Hilbert delivers his address to the International Congress of Mathematicians. Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented only ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the Second International Congress of Mathematicians, speaking on 8 August in the Sorbonne.
"Who of us would not be glad to lift the veil behind which the future is hidden..."
He had provided an extract of the speech (most unusual at that time) in French(?English?) before hand for those who were not fluent in French.



1931 George Birkhoff publishes “A Set of Postulates for Plane Geometry Based on Scale and Protractor" in Annals of Mathematics. The system has undefined elements of point and line, and undefined relations of distance and angle. (pb)
in 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms. Birkhoff's axiom system was utilized in the secondary-school text Basic Geometry (first edition, 1940) *Wik



1975 The term global warming was probably first used in its modern sense on 8 August 1975 in a science paper by Wally Broecker in the journal Science called "Are we on the brink of a pronounced global warming?". Broecker's choice of words was new and represented a significant recognition that the climate was warming; previously the phrasing used by scientists was "inadvertent climate modification," because while it was recognized humans could change the climate, no one was sure which direction it was going. The National Academy of Sciences first used global warming in a 1979 paper called the Charney Report, which said: "if carbon dioxide continues to increase, [we find] no reason to doubt that climate changes will result and no reason to believe that these changes will be negligible." The report made a distinction between referring to surface temperature changes as global warming, while referring to other changes caused by increased CO2 as climate change.

Global warming became more widely popular after 23 June, 1988 when NASA climate scientist James Hansen used the term in a testimony to Congress. He said: "global warming has reached a level such that we can ascribe with a high degree of confidence a cause and effect relationship between the greenhouse effect and the observed warming." His testimony was widely reported and afterward global warming was commonly used by the press and in public discourse. *Wik




1977 Derek T. Whiteside received the Sarton Medal, the highest honor that the History of Science Society can bestow, for his editorship of The Mathematical Papers of Isaac Newton. In delivering the award Richard S. Westfall said “Before Tom began, Newton’s mathematics was largely a land of myth and fable.” In his 25 years work on the papers, Whiteside has changed all that. *VFR 

An account of Newton’s discovery of universal gravitation in John Conduitt’s hand. Conduitt was Newton’s assistant at the Royal Mint and married his niece Catherine  Barton.

From manuscript evidence, historians date it to around 1726, when Conduitt was gathering material for a projected biography of Newton. The account survives in his own hand among the Portsmouth Papers (now in the University of Cambridge Library, MS Keynes 130/4), and it was never published in his lifetime. It’s the source for the famous “apple falling in the garden” anecdote that later biographers such as William Stukeley and Voltaire popularized. *PBnotes



2008 The wreckage of the Hunley, a Confederate submarine that was lost during the American Civil War, was raised from the ocean floor near Sullivans Island, South Carolina; it was the first submarine to sink (1864) an enemy ship (the Union sloop Housatonic).
The Hunley was designed and built at Mobile, Alabama, and named for its chief financial backer, Horace L. Hunley. Less than 40 feet (12 metres) long, the submarine could hold up to nine crewmen, most of whom propelled the vessel by hand cranking a single screw. Its commander controlled steering and depth. The Hunley was shipped by rail in 1863 to Charleston, South Carolina, where it was launched in July. In practice runs and attempts to attack blockading Union warships, it went to the bottom three times with great loss of life—including that of Hunley himself. Raised one more time, it successfully attacked the Union sloop Housatonic with a spar torpedo on February 17, 1864, sinking the vessel. The Hunley, however, was lost shortly after the attack, along with its eight crewmen.

Byname: Hunley
The vessel lay in only 30 feet (9 metres) of water some 4 miles (6 km) offshore until it was found by preservationists in 1995. It was raised intact in 2000 and taken to North Charleston’s Warren Lasch Conservation Center, which had been constructed for the Hunley. The crewmen’s remains were later removed for burial, and the submarine underwent extensive preservation work and research. Of particular interest was the cause of the crew’s death, long thought to be suffocation or drowning. However, when the Hunley was unsealed, the bodies were found at their posts, and there was no indication that the men had tried to evacuate. In addition, the submarine showed no major damage. Various theories were proposed, and in 2017 researchers at Duke University speculated that the blast from the torpedo that sunk the Housatonic produced a shock wave that ruptured blood vessels in the men’s lungs. Known as blast lung, it either killed the crew instantaneously or incapacitated them, causing the Hunley to sink.*Britannica
Maybe if they had waited,.  As I'm writing this in 2026, Europe is in a major drought The 2nd longest river in Europe is so dry that along parts of the Danube, WWII German battleships are appearing in the river where they had been sunk by the Germans ahead of the oncoming Russian advance,C'est la Vie!


***Hunley, 19th-century illustration.




In 2007, Barbara Morgan became the first educator to safely reach space was launched on the U.S. Space Shuttle Endeavour to the International Space Station. In 1986, she was the alternate for the first teacher selected for a space mission, Christa McAuliffe (who died with six astronauts in the explosion of the Challenger space shuttle 73-sec after its launch). When the Endeavour reached orbit, Mission Control announced: “For Barbara Morgan and her crewmates, class is in session.” During the flight, Morgan spoke with students in Idaho, where she had taught elementary classes. She had astronaut training in Houston (1998) Since its previous flight in 2002, the Endeavour had a massive overhaul.*TiS




2013 Duck
!!! Oops too late, The new radar images show the asteroid 2005 WK4 as it passed Earth at a safe distance of 1.93 million miles (3.1 million kilometers), which is about 8.2 times the distance between Earth and the moon. The images revealed the large asteroid to be between 660 and 980 feet across (200 to 300 meters), *NASA


BIRTHS


1627 Joseph Moxon (8 August 1627 - February 1691 (Royal Society archives state his death date as 28 February; the Oxford Dictionary of National Biography states that he was buried on 15 February???{I hope one of them was wrong}), hydrographer to Charles II, was an English printer of mathematical books and maps, a maker of globes and mathematical instruments, and mathematical lexicographer. He produced the first English language dictionary devoted to mathematics, "Mathematicks made easie, or a mathematical dictionary, explaining the terms of art and difficult phrases used in arithmetick, geometry, astronomy, astrology, and other mathematical sciences". In November 1678, he became the first tradesman to be elected as a Fellow of the Royal Society. *Wik Thony Christie has written that he was one of the first English Printers to print tables of Logarithms.


1901 Ernest Orlando Lawrence (August 8, 1901 – August 27, 1958) American physicist who was awarded the 1939 Nobel Prize for Physics for his invention of the cyclotron, the first device for the production of high energy particles. His first device, built in 1930 used a 10-cm magnet. He accelerated particles within a cyclinder at high vacuum between the poles of an electromagnetic to confine the beam to a spiral path while a high A.C. voltage increased the particle energy. Larger models built later created 8 x 104 eV beams. By colliding particles with atomic nuclei, he produced new elements and artificial radioactivity. By 1940, he had created plutonium and neptunium. He extended the use of atomic radiation into the fields of biology and medicine. Element 103 was named Lawrencium as a tribute to him.*TIS
The first cyclotron was 4 ½ inches across, so it is sometimes referred to as the 4” cyclotron, sometimes as the 5”, but it is all the same instrument.  In the images above we see Lawrence holding the 4” cyclotron 

*Linda Hall Org



1902 Paul Adrien Maurice Dirac, OM, FRS (8 August 1902 – 20 October 1984) English theoretical physicist known for his work in quantum mechanics and for his theory of the spinning electron. In 1933 he shared the Nobel Prize for Physics with the Austrian physicist Erwin Schrödinger. *TIS One of my favorite Dirac anecdotes (of which there are many)
Dirac was watching Anya Kapitza knitting while he was talking physics with Peter Kapitza. A couple of hours after he left, Dirac rushed back, very excited. "You know, Anya," he said, "watching the way you were making this sweater I got interested in the topological aspect of the problem. I found that there is another way of doing it and that there are only two possible ways. One is the one you were using; another is like that. . . . " And he demonstrated the other way, using his long fingers. His newly discovered "other way," Anya informed him, is well known to women and is none other than "purling."

Douglas W Boone commented that, "Many discoveries are re-discoveries.  But Dirac's naivety (supposing that purling was an innovation) is charming."

"On the other hand, he confirmed that knitting and purling are a closed set: he proved that another stitch will not be found."

Dirac is buried in Tallahassee, Florida, but a memorial plaque has been installed in Westminster Abbey, sporting the Dirac equation . The move to install the plaque drew some understandable protest from the Dean of the Abbey, since Dirac was an atheist. But he is far from being the only non-believer to be honored within its walls; Charles Darwin is actually buried there, not too far away.




1910 Johanna Weber (8 August 1910 – 24 October 2014) was a German-born British mathematician and aerodynamicist. She is best known for her contributions to the development of the Handley Page Victor bomber and the Concorde.

In 1939, Weber joined the Aerodynamics Research Institute (Aerodynamische Versuchsanstalt Göttingen) in Göttingen. She was part of a small theoretical team, and her initial training in aerodynamics consisted of wind tunnel corrections. Here she met and began her lifelong collaboration with Dietrich Küchemann.

Scientists at Institute had by then worked out a consistent theory of flow around an aircraft. This was, however, an approximation, using singularities to represent the vortices that generated lift, and Weber was given the task of improving it. She realised that some of her work overlapped with Küchemann's research on jet engine intakes. They teamed up, with Weber doing the theoretical development and wind tunnel testing, and Küchemann setting the direction of their research based on his consultation with manufacturers. Over the period of the Second World War, they created a substantial body of work.

Weber also began her research into supersonic transport. In 1955, she showed that a thin delta wing with a high angle of attack could generate sufficient lift to provide the take-off and landing capability, while simultaneously enabling efficient supersonic performance. Küchemann then advocated this wing configuration with the UK Government, resulting in the support for a Mach 2 airliner by the Supersonic Transport Advisory Committee (STAC) in 1956.

In 1961, a prototype aircraft, the Handley Page HP.115, was built to test the low speed performance of the slender delta wing.*Wik




1921 Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, and wrote what was for a long time the standard textbook on algebraic topology (Spanier 1981).


Spanier attended the University of Minnesota, graduating in 1941. During World War II, he served in the United States Army Signal Corps. He received his Ph.D. degree from the University of Michigan in 1947 for the thesis Cohomology Theory for General Spaces written under the direction of Norman Steenrod. After spending a year as a research fellow at the Institute for Advanced Study in Princeton, New Jersey, in 1948 he was appointed to the faculty of the University of Chicago, and then a professor at UC Berkeley in 1959. He had 17 doctoral students, including Morris Hirsch and Elon Lages Lima.



1931 Sir Roger Penrose, British mathematician and theoretical physicist who in the 1960s calculated many of the basic features of black holes.*TIS A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.*Wik

Roger’s siblings all became intellectuals set on cracking their own conundrums. His older brother Oliver became a highly respected professor of statistical mechanics, exploring the mysteries of liquid helium and Bose-Einstein condensates. His younger brother Jonathan was a chess prodigy, winning the British Chess Championship a record 10 times, while his sister Shirley became a distinguished geneticist.



1952 Roman Juszkiewicz (born 8 August 1952, died 28 January 2012) is a Polish astrophysicist whose work is concerned with fundamental issues of cosmology.
Juszkiewicz's scientific interests include the theory of gravitational instability, origins of the large-scale structure, microwave background radiation and Big Bang nucleosynthesis. He wrote nearly one hundred research papers, mostly in the area of cosmology. Calculated results based on observed motions of pairs of galaxies, obtained in 2000 by Roman Juszkiewicz and the group led by him, aimed at estimating the amount of dark matter in the Universe, were confirmed by the recently published data from the South Pole's ACBAR detector. *Wik






DEATHS

1555 Oronce Fine (20 December 1494 - 8 August 1555) was a French mathematician who published a major work on mathematics and astronomy.*SAU Although primarily a popularizer, Fine was one of the most prolific authors of mathematical books of his age. He worked in a wide range of mathematical fields, including practical geometry, arithmetic, optics, gnomonics, astronomy, and instrumentalism.
He gave the value of pi to be (22 2/9)/7 in 1544. Later, he gave 47/15 and, in De rebus mathematicis (1556), he gave 3 11/78. *Wik



1979 Jacob Lionel Bakst Cooper (27 December 1915 – 8 August 1979) was a South African mathematician who worked in operator theory, transform theory, thermodynamics, functional analysis and differential equations.1979  
Within operator theory, Cooper worked in the area of linear operators on real or complex Hilbert spaces. He studied the unbounded operators that arose from quantum theory, extending basic work of Frigyes Riesz and John von Neumann.

Within transform theory, he worked on the representation and uniqueness of integral transforms, on approximation, and on linear transformations that satisfy functional relations arising from representations of linear groups In this he collaborated closely with P.L. Butzer of RWTH Aachen. *Wik



1989 Taro Morishima (22 April 1903 in Wakayama, Japan - 8 Aug 1989 in Tokyo, Japan)...His passion was algebraic number theory and he had a particular love of Fermat's Last Theorem. His first paper on Fermat's Last Theorem was published in the Proceedings of the Imperial Academy of Japan in 1928. It was the first of 12 papers written in German with the title Über die Fermatsche Vermutung, with 10 of these papers being in the Proceedings of the Imperial Academy of Japan between the years 1928 and 1935. By 1935 he had published a total of 16 papers, the other 6 being: Über den Fermatschen Quotienten (1931); On recent results about Fermat's last Theorem (Japanese) (1932); Über die Einheiten und Idealklassen des Galoisschen Zahlkörpers und die Theorie der Kreiskörper der l-ten Einheitswurzein (1933); and Über die Theorie der Kreiskörper der l-ten Einheitswurzein (1935). He also published a monograph Fermat's Problem (1934) in Japanese. All his papers are full of good ideas but they are extremely difficult to read since Morishima did not present enough detail.
Morishima's high research activity seems to have greatly lessened after 1935. Although difficulties relating to World War II​ and the difficult years in Japan following the war were partly responsible, nevertheless it does appear that he had already reduced his research activities. He did publish the book Higher Algebra in 1940 (in Japanese) but this and one further paper on Fermat's Last Theorem (in 1952) was all in published in the 30 years between 1935 and 1965.*SAU



1996 Sir Nevill F(rancis) Mott (30 September 1905 – 8 August 1996) English physicist who shared (with P.W. Anderson and J.H. Van Vleck of the U.S.) the 1977 Nobel Prize for Physics for his independent researches on the magnetic and electrical properties of amorphous semiconductors. Whereas the electric properties of crystals are described by the Band Theory - which compares the conductivity of metals, semiconductors, and insulators - a famous exception is provided by nickel oxide. According to band theory, nickel oxide ought to be a metallic conductor but in reality is an insulator. Mott refined the theory to include electron-electron interaction and explained so-called Mott transitions, by which some metals become insulators as the electron density decreases by separating the atoms from each other in some convenient way.*TIS



2009 Muriel Kennett Wales (9 Jun 1913 – 8 August 2009) was an Irish-Canadian mathematician, and is believed to have been the first Irish-born woman to earn a PhD in pure mathematics.
She was first educated at the University of British Columbia (BA 1934, MA 1937 with the thesis Determination of Bases for Certain Quartic Number Fields). In 1941 she was awarded the PhD from the University of Toronto for the dissertation Theory Of Algebraic Functions Based On The Use Of Cycles under Samuel Beatty  (himself the first person to receive a PhD in mathematics in Canada, in 1915).

She spent most of the 1940s working in atomic energy, in Toronto and Montreal, but by 1949 had retired back to Vancouver where she worked in her step-father's shipping company.*Wik



2017 Cathleen Synge Morawetz (May 5, 1923; Toronto, Canada - August 8, 2017 ) is a mathematician. Morawetz's research was mainly in the study of the partial differential equations governing fluid flow, particularly those of mixed type occurring in transonic flow. She is Professor Emerita at the Courant Institute of Mathematical Sciences at the New York University, where she has also served as director from 1984 to 1988.
Morawetz's father, John Lighton Synge was an Irish mathematician, specializing in the geometry of general relativity and her mother also studied mathematics for a time. Her childhood was split between Ireland and Canada. Both her parents were supportive of her interest in mathematics and science, and it was a woman mathematician, Cecilia Krieger, who had been a family friend for many years who later encouraged Morawetz to pursue a PhD in mathematics. Morawetz says her father was influential in stimulating her interest in mathematics, but he wondered whether her studying mathematics would be wise (suggesting they might fight like the Bernoulli brothers)
In 1981, she became the first woman to deliver the Gibbs Lecture of The American Mathematical Society, and in 1982 presented an Invited Address at a meeting of the Society for Industrial and Applied Mathematics. She was named Outstanding Woman Scientist for 1993 by the Association for Women in Science. In 1995, she became the second woman elected to the office of president of the American Mathematical Society. In 1998 she was awarded the National Medal of Science; she was the first woman to receive the medal for work in mathematics. In 2004 she received the Leroy P. Steele Prize for Lifetime Achievement. In 2006 she won the George David Birkhoff Prize in Applied Mathematics. In 2012 she became a fellow of the American Mathematical Society.*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