Pat'sBlog

Saturday, 20 June 2026

On This Day in Math - June 20

the enormous success of mathematics in the natural
sciences is something bordering on the mysterious and ...
there is no natural explanation for it.

—Eugene Wigner

The 171st day of the year; 171 has the same number of digits in Roman numerals as its cube.
CLXXI^3 =\( \overline{V}CCXI \) 5000211

\( 10^{171 } - 171 \) is a prime number with 168 9's followed by 829

Google calculator gives 171! = infinity. (close enough in many cases)

171 is the 18th triangular number and is the last year-day that is both a triangular number and a palindrome. *Ben Vitale

EVENTS

1667 Louis XIV, The Sun King of France, attends the ceremony of inauguration of the Observatoire de Paris, the oldest working observatory in the world. *Amir D. Aczel, Pendulum, pg 66

1686 Halley Writes to Newton that Hooke has protested his "discovery" of the inverse square law should be noted in Principia. Newton will respond On July 14, 1686, with a peace offering; "And now having sincerely told you the case between Mr Hooke and me, I hope I shall be free for the future from the prejudice of his letters. I have considered how best to compose the present dispute, and I think it may be done by the inclosed scholium to the fourth proposition." This scholium was "The inverse law of gravity holds in all the celestial motions, as was discovered also independently by my countrymen Wren, Hooke and Halley."

1688 Newton, in a letter to Edmund Halley, again expresses his exasperation with carping critics. [Thanks to Howard Eves]*VFR

1788; Washington Writes to Nicholas Pike to Thank him for a copy of his "A New and Complete System of Arithmetic" , published in 1786 by Nicholas Pike, a Newburyport schoolmaster. In his letter, sent June 20, 1788, from Mount Vernon, Washington writes: "The handsome manner in which that Work is printed and the elegant manner in which it is bound, are pleasing proofs of the progress which the Arts are making in this Country. Washington's letter to Pike also commended him on his accomplishments and the importance of his work.
Pike had written to Washington on March 25,1786 requesting permission to dedicate the book to Washington. On June 20 of 1786, Washington had replied that, "I must therefore beg leave to decline the honour which you would do me, as I have before done in two or three cases of a similar kind."

1808 Poisson submitted his ﬁrst paper on the stability of the planetary system, one day before his twenty-seventh birthday. *VFR

His memoirs on celestial mechanics," in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important of these are his memoirs Sur les inégalités séculaires des moyens mouvements des planètes." In this memoir, Poisson discusses the famous question of the stability of the planetary orbits, which had already been settled by Lagrange to the first degree of approximation for the disturbing forces. Poisson showed that the result could be extended to a second approximation, and thus made an important advance in planetary theory. The memoir is remarkable inasmuch as it roused Lagrange, after an interval of inactivity, to compose in his old age one of the greatest of his memoirs, entitled Sur la théorie des variations des éléments des planètes, et en particulier des variations des grands axes de leurs orbites. So highly did he think of Poisson's memoir that he made a copy of it with his own hand, which was found among his papers after his death. Poisson made important contributions to the theory of attraction. *Wik

1831 János Bolyai's pioneering work, The Absolutely True Science of Space, was published in 1832. This important work was published as an appendix to the first volume of his father,Farkas Bolyai's Tentamen , but its off-print had already been ready the previous year, in April 1831. The latter was the version which, together with a letter, was sent to Gauss by Farkas Bolyai on the 20th of June 1831. Gauss got the letter but János's work was lost on the way. On the 16th of January 1832 Farkas sent the Appendix to his friend again with another letter in which he wrote: ``My son appreciates Your critique more than that of whole Europe and it is the only thing he is waiting for''.
After twenty-three years of silence, Gauss replied to his ``old, unforgettable friend'' on the 6th of March 1832. One of his well-known sentences was: ``if I praised your son's work I would praise myself''. The letter deeply afflicted and upset János Bolyai, although it reflects appreciation, too: ``... I am very glad that it is my old friend's son who so splendidly preceded me'' *Komal Journal

1877 Georg Cantor, in a letter to Dedekind, announced a proof that the points inside a square are in one-to-one correspondence with those on a line segment. Three years earlier, Cantor had intimated that this was clearly impossible. *VFR

A part of the letter, with a HatTip to Offer Pade'.

1908 Count Zeppelin made his first flight in his fourth new airship at Friedrichshafen, Germany. The Luftschiff LZ4 had its first flight 20 Jun 1908. Its first extended flight (12 hours) was taken to Switzerland 1 Jul 1908. At the beginning of August, it embarked on an extended flight which had taken it among other places to Basel, Straussberg, and many of the major cities of southern Germany. While moored at Echterdingen on 5 Aug 1908, it was torn from the mast by high winds and destroyed. As interest in the Zeppelins ran high in German, the incident was felt as a national disaster. Spontaneous donations resulted in approximately 5.5 million Marks and made it possible for Zeppelin to continue his work. *TIS

*Wik

In 1979, 32 solar panels on the White House roof, installed by the Carter administration, were dedicated. President Carter wished to demonstrate a committment to renewable energy use, as a model for the nation. About 3m x 1m x 10cm deep, the dark surfaces of the panels absorbed energy from sunlight, heating water passing through pipes snaking below them. Carter stated, “In the year 2000 this solar water heater behind me, which is being dedicated today, will still be here supplying cheap, efficient energy.” That was not to be. The subsequent Republican President, Ronald Reagan, with one of the worst environmental records of any president, while the roof was being resurfaced in 1986, had them removed and sent to warehouse storage. In the same year he slashed the research and development budget for renewable energy, and eliminated tax breaks for wind turbines and solar projects. *TiS

BIRTHS

1775 Jacques Frédéric Français (20 June 1775 in Saverne, Bas-Rhin, France - 9 March 1833 in Metz, France) In September 1813 Français published a work in which he gave a geometric representation of complex numbers with interesting applications. This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to François Français. Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians. This changed after Français' paper since a vigorous discussion between Français, Argand and Servois took place in Gergonne's Journal. In this argument Français and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra. *SAU

1838 Theodor Reye (20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) worked in Geometry and Projective Geometry.*SAU

He is best known for his introduction of configurations in the second edition of his book, Geometrie der Lage (Geometry of Position, 1876). The Reye configuration of 12 points, 12 planes, and 16 lines is named after him.

Reye also developed a novel solution to the following three-dimensional extension of the problem of Apollonius: Construct all possible spheres that are simultaneously tangent to four given spheres.

1873 Alfred Loewy born.(20 June 1873 in Rawitsch, Germany (now Rawicz, Poznań, Poland) - 25 Jan 1935 in Freiburg im Breisgau, Germany) He worked in group theory and differential equations. *VFR

Loewy was appointed as an extraordinary professor at Freiburg in 1902. This made him secure enough financially to marry and in that year he married Therese Neuburger. Loewy became an honorary ordinary professor at Freiburg in 1916 before his appointment as ordinary professor in 1919. He was thesis advisor to a number of famous students, in particular Wolfgang Krull, who was awarded his doctorate in 1922, and Friedrich Karl Schmidt, who was awarded his doctorate in 1925. Other algebraists who spent some time in Freiburg working under Loewy are E Witt, Bernhard Neumann, R Brauer, R Baer, and A Scholz.

Anti-Semitism increased in Germany following the end of World War I. Anti-Semites joined forces with nationalists in attempting to blame the Jews for Germany's defeat. Increasing discrimination was not the only source of difficulty in Loewy's life. Already by 1916 he had lost the sight of one eye. His eyesight began to fail completely from about 1920 and he became totally blind before his death after a failed operation in 1928 left his other eye completely blind also. Despite these severe health problems Loewy continued to carry out his teaching duties. He could battle against blindness and against the hurt of anti-Semitism directed at him, but the final blow came in 1933 when anti-Semitism became part of the law of the land. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law was passed that provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Loewy was forced to retire in 1933 under the Civil Service Law.

Among Loewy's most famous books are Lehrbuch der Algebra Ⓣ (1915) and Mathematik des Geld- und Zahlungsverkehrs Ⓣ (1920). The first of these was one of the first works to introduce into Germany the methodology, the terminology and the achievements of postulational analysis as it was being developed in the United States.

Loewy was Abraham Fraenkel's uncle by marriage and he exerted a large influence on Fraenkel's career in its early stages. It was Loewy who persuaded Fraenkel to travel to Marburg to study under Hensel and it was Loewy who had helped Fraenkel publish his early work in Crelle's journal with a paper about the date of Easter. But the mathematical topics Fraenkel studied were also influenced by Loewy whose interest in the study of axiomatic systems encouraged a similar interest by Fraenkel. The relationship worked both ways round, however, and Loewy's Grundlagen der Arithmetik Ⓣ, published in 1915, was prepared with Fraenkel's assistance. Loewy mentioned in this work that in the system of integers, the product of any two integers is zero, if and only if one of them is zero. Such ideas clearly influenced Fraenkel to introduce the notion of a ring, and in particular zero-divisors in rings. *SAU

1917 Helena Rasiowa (20 June 1917 – 9 August 1994) was a Polish mathematician. She worked in the foundations of mathematics and algebraic logic.

There was an impressive collection of mathematicians at the University of Warsaw at this time including Borsuk, Łukasiewicz, Mazurkiewicz, Sierpiński, Mostowski and others. They had organised an underground version of the university which was strongly opposed by the Nazi authorities. Borsuk, for example, was imprisoned after the authorities found that he was helping to run the underground university.

In this dangerous situation Rasiowa learnt mathematics, knowing that the penalties for being discovered were extreme. Yet in this environment Rasiowa studied for her Master's Degree under Łukasiewicz's supervision.

In 1946, having obtained her Master's degree, she was appointed as an assistant at the University of Warsaw and continued to work for her doctorate under Mostowski's supervision. Her thesis, presented in 1950, was on algebra and logic Algebraic treatment of the functional calculus of Lewis and Heyting and these topics would be the main areas of her research throughout her life.

Rasiowa was promoted steadily, reaching the rank of Professor in 1957 and Full Professor in 1967. She led the Foundations of Mathematics Section from 1964 and the Mathematical Logic Section after its creation in 1970.

Her main research was in algebraic logic and the mathematical foundations of computer science. In algebraic logic she continued work by Post, Stone, Tarski and Łukasiewicz :-

... aimed at finding a precise description for the mathematical structure of formalised logical systems.

Of course Rasiowa's work on algebraic logic was in precisely the right area to make her a natural contributor to theoretical computer science. However it is one thing to be in the right area and yet another to have the ability to see the importance of a new subject such as computer science. Her contributions are described in :-

Her contribution to theoretical computer science stems from her conviction that there are deep relations between methods of algebra and logic on the one side and essential problems of foundations of computer science on the other. Among these problems she clearly distinguished inference methods characteristic of computer science and its applications. This conviction of hers had been supported by her results on many-valued and non-classical logics, especially on applications of various generalisations of Post algebras to logics of programs and approximation logics.

In fact in 1984 Rasiowa introduced an important concept of inference where the basic information was incomplete. This led to approximate reasoning and approximate logics which are now central to the study of artificial intelligence. *SAU

1940 Leonard Susskind ( June(20ish) 1940 - )(The professor's real birthday seems difficult to determine; perhaps only known to him and his parents, perhaps only to his parents) is the Felix Bloch Professor of Theoretical Physics at Stanford University, and Director of the Stanford Institute for Theoretical Physics. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology. He is a member of the National Academy of Sciences, and the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics, and a distinguished professor of the Korea Institute for Advanced Study.
Susskind is widely regarded as one of the fathers of string theory, having, with Yoichiro Nambu and Holger Bech Nielsen, independently introduced the idea that particles could in fact be states of excitation of a relativistic string. He was the first to introduce the idea of the string theory landscape in 2003. *Wik

1942 Neil Sidney Trudinger (20 June 1942 - ) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations.

At the ANU Trudinger served as Head of the Department of Pure Mathematics, as Director of the Centre for Mathematical Analysis and as Director of the Centre for Mathematics and its Applications, before becoming Dean of the School of Mathematical Sciences in 1992. He currently coordinates ANU's Applied and Nonlinear Analysis program. He is co-author, together with his thesis advisor, David Gilbarg, of the book Elliptic Partial Differential Equations of Second Order.

His long list of awads includes :

2008, awarded the Leroy P. Steele Prize for Mathematical Exposition by the American Mathematical Society.

2012, elected as a fellow of the American Mathematical Society.

2014, gave the Łojasiewicz Lecture (on the "Optimal Transportation in the 21st Century") at the Jagiellonian University in Kraków. *Wik

1946 Nigel John Kalton (June 20, 1946 – August 31, 2010) was a British-American mathematician, known for his contributions to functional analysis.

After studying mathematics at Trinity College, Cambridge, he received his PhD, which was awarded the Rayleigh Prize for research excellence, from Cambridge University in 1970. He then held positions at Lehigh University in Pennsylvania, Warwick, Swansea, University of Illinois, and Michigan State University, before becoming full professor at the University of Missouri, Columbia, in 1979.

He received the Stefan Banach Medal from the Polish Academy of Sciences in 2005] A conference in honour of his 60th birthday was held in Miami University of Ohio in 2006. He died in Columbia, Missouri, aged 64.

David Kazhdan (Hebrew: דוד קשדן), born Dmitry Aleksandrovich Kazhdan (20 June 1946 ), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 MacArthur Fellow.

In 2002, he immigrated to Israel and is now a professor at the Hebrew University of Jerusalem as well as a professor emeritus at Harvard. Perhaps the most famous of the students that Kazhdan advised for a Ph.D. at Harvard was Vladimir Voevodsky who was awarded a Ph.D. in 1992 for his thesis Homology of Schemes and Covariant Motives. The ideas in this thesis eventually led to work which saw Voevodsky awarded a Fields Medal in 2002.

On October 6, 2013, Kazhdan was critically injured in a car accident while riding a bicycle in Jerusalem.

DEATHS

1800 Abraham Kästner (27 September 1719 – 20 June 1800) was a German mathematician who compiled encyclopaedias and wrote text-books. He taught Gauss. *SAU

He was known in his professional life for writing textbooks and compiling encyclopedias rather than for original research. Georg Christoph Lichtenberg was one of his doctoral students, and admired the man greatly. He became most well-known for his epigrammatic poems. The crater Kästner on the Moon is named after him.

An epigram is a brief, interesting, memorable, sometimes surprising or satirical statement. This literary device has been practiced for over two millennia. The presence of wit or sarcasm tends to distinguish non-poetic epigrams from aphorisms and adages, which typically do not show those qualities.

What is an Epigram? a dwarfish whole,

Its body brevity, and wit its soul.

— Samuel Taylor Coleridge ("Epigram", 1809)

Here lies my wife: here let her lie!

Now she's at rest – and so am I.

— John Dryden

*Wik

1807 Ferdinand Berthoud (19 March 1727 – 20 June 1807) Outstanding Swiss horologist and author of extensive treatises on timekeeping who became involved in the attempt to solve the problem of determining longitude at sea. His major achievement was his further development of an accurate and practical marine clock, or chronometer. (Such an instrument had previously been constructed in expensive and delicate prototypes by Pierre Leroy of France and John Harrison of England.) He made his first chronometer in 1754, which was sent for trial in 1761. Berthoud's improvements to the chronometer have been largely retained in present-day designs. *TIS

1861 Sir Frederick Gowland (Hoppy) Hopkins OM PRS (20 June 1861 – 16 May 1947) was an English biochemist who was awarded the Nobel Prize in Physiology or Medicine in 1929, with Christiaan Eijkman, for the discovery of vitamins. He also discovered the amino acid tryptophan, in 1901. He was President of the Royal Society from 1930 to 1935. His Cambridge students included neurochemistry pioneer Judah Hirsch Quastel and pioneer embryologist Joseph Needham.
During his life, in addition to the Nobel Prize, Hopkins was awarded the Royal Medal of the Royal Society in 1918 and the Copley Medal of the Royal Society in 1926. Other significant honours were his election in 1905 to fellowship in the Royal Society, Great Britain's most prestigious scientific organisation; his knighthood by King George V in 1925; and the award in 1935 of the Order of Merit, Great Britain's most exclusive civilian honour. From 1930 -1935 he served as president of the Royal Society and in 1933 served as President of the British Association for the Advancement of Science. *Wik

1865 Sir John William Lubbock, (London, England, 26 March 1803 - Downe, Kent, England, 20 June 1865 )English astronomer and mathematician. He made a special study of tides and of the lunar theory and developed a method for calculating the orbits of comets and planets. In mathematics he applied the theory of probability to life insurance problems. He was a strong proponent of Continental mathematics and astronomy.
Lubbock, third Baron Lubbock, was born into a London banking family. After attending Eton, he moved to Trinity College, Cambridge, where he became a student of William Whewell.(it was at the request of Lubbock that Whewell created the term "biometry".) He excelled in mathematics and traveled to France and Italy to deepen his knowledge of the works of Pierre-Simon de Laplace and Joseph Lagrange. Entering his father’s banking firm as a junior partner, he devoted his free time to science.
Lubbock strongly supported Lord Brougham’s Society for the Diffusion of Useful Knowledge [SDUK], which produced scientific and technical works designed for the working class. His articles on tides for the Society’s publications resulted in a book, *An Elementary Treatise on the Tides, in 1839. *Biographical Encyclopedia of Astronomers

1963 Raphaël Salem (November 7, 1898 in Salonika, Ottoman Empire (now Thessaloniki, Greece) – June 20, 1963 in Paris, France) was a Greek mathematician after whom are named the Salem numbers and Salem–Spencer sets, and whose widow founded the Salem Prize.

Salem left England in the autumn of 1940 and emigrated to the United States where he settled in Cambridge, Massachusetts. In 1941, he was appointed as a lecturer in mathematics at MIT, where he was rapidly promoted and became an assistant and associate professor. In 1958, he was appointed as Professor at the Sorbonne and lived in Paris until his death in 1963. In 1967, Éditions Hermann published Salem's Oeuvres mathématiques, edited by his collaborators Antoni Zygmund and Jean-Pierre Kahane. After Salem's death, his widow established the Salem Prize, an international prize given to young researchers for outstanding contributions to Fourier series.

In mathematics, a Salem number is a real algebraic integer 𝛼>1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem. Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be a reciprocal polynomial. This implies that 1/𝛼 is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a unit in the ring of algebraic integers, being of norm 1.

1966 Georges (Henri) 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.

1981 Henri-Gaston Busignies (29 Dec 1905; 20 Jun 1981) French-born American electronics engineer whose invention (1936) of high-frequency direction finders (HF/DF, or "Huff Duff") permitted the U.S. Navy during World War II to detect enemy transmissions and quickly pinpoint the direction from which a radio transmission was coming. Busignies invented the radiocompass (1926) while still a student at Jules Ferry College in Versailles, France. In 1934, he started developing the direction finder based on his earlier radiocompass. Busignies developed the moving target indicator for wartime radar. It scrubbed off the radar screen every echo from stationary objects and left only echoes from moving objects, such as aircraft. *TIS

1990 Kōsaku Yosida ( 7 February 1909, Hiroshima – 20 June 1990) was a Japanese mathematician who worked in the field of functional analysis. He is known for the Hille-Yosida theorem concerning C0-semigroups. Yosida studied mathematics at the University of Tokyo, and held posts at Osaka and Nagoya Universities. In 1955, Yosida returned to the University of Tokyo. *Wik

In 1933 Yosida was appointed as an Assistant in the Department of Mathematics at Osaka Imperial University. Osaka is on Honshu Island, roughly half way between Hiroshima and Tokyo. The Osaka Imperial University is based on educational institutions dating back to the 18th century but only became a university in 1931, two years before Yosida was appointed there. After one year, he was promoted to Associate Professor.

Moving to Osaka Imperial University led to Yosida changing the direction of his research. Two mathematicians who joined the Department of Mathematics shortly after him and were to strongly influence him were Mitio Nagumo (1905-1995) and Shizuo Kakutani. Nagumo had graduated from Tokyo Imperial University in March 1928 and spent two years at the University of Göttingen, Germany, before being appointed to Osaka Imperial University in March 1934. Kakutani had studied at Tohoku University in Sendai before being appointed as a teaching assistant at Osaka Imperial University in 1934. Yosida became interested in functional analysis through discussions with these two mathematicians. He published several joint papers with Kakutani. *SAU

2003 I. Bernard Cohen (1 March 1914 – 20 June 2003) was the Victor S. Thomas Professor of the history of science at Harvard University and the author of many books on the history of science and, in particular, Isaac Newton.
Cohen was the first American to receive a Ph.D. in history of science, was a Harvard undergraduate ('37) and then a Ph.D. student and protégé of George Sarton who was the founder of Isis and the History of Science Society. Cohen taught at Harvard from 1942 until his death, and his tenure was marked by the development of Harvard's program in the history of science. *Wik

2005 Jack St. Clair Kilby (8 November 1923 - 20 June 2005) was an American electrical engineer who took part, along with Robert Noyce of Fairchild Semiconductor, in the realization of the first integrated circuit while working at Texas Instruments (TI) in 1958. He was awarded the Nobel Prize in Physics on 10 December 2000.

Kilby was also the co-inventor of the handheld calculator and the thermal printer, for which he had the patents. He also had patents for seven other inventions. *Wik

Jack Kilby's original integrated circuit

2012 William Wager Cooper (July 23, 1914 – June 20, 2012) was an American operations researcher, known as a father of management science and as "Mr. Linear Programming". He was the founding president of The Institute of Management Sciences, founding editor-in-chief of Auditing: A Journal of Practice and Theory, a founding faculty member of the Graduate School of Industrial Administration at the Carnegie Institute of Technology (now the Tepper School of Business at Carnegie Mellon University), founding dean of the School of Urban and Public Affairs (now the Heinz College) at CMU, the former Arthur Lowes Dickinson Professor of Accounting at Harvard University, and the Foster Parker Professor Emeritus of Management, Finance and Accounting at the University of Texas at Austin.

Cooper was born in Birmingham, Alabama and grew up in Chicago, where his father (a former bookkeeper) owned several gasoline stations that closed in the Great Depression. Cooper, in his second year of high school, dropped out to help support his family. He worked in a bowling alley, on a golf course, and as a professional boxer. As a boxer, he won 58 bouts, lost three, and drew two. While commuting to the golf course, he met Eric Kohler, a professor at Northwestern University, who pushed him to go back to school and bankrolled his entry to the University of Chicago. At Chicago, he began studying physical chemistry but was inspired by his work for Kohler on a legal case to switch to economics, graduating with a B.A. and Phi Beta Kappa honors in 1938.

After graduation, from 1938 to 1940, he worked as an accountant for the Tennessee Valley Authority, where Kohler had become Controller. There, he worked on performance auditing and the mathematical allocation of resources, and helped Kohler testify to a congressional investigative committee. In 1940, Cooper started doing graduate studies at Columbia University; however, in 1942, with his coursework completed but his thesis unwritten, he left Columbia to serve his country during World War II. He worked in the Division of Statistical Standards of the U.S. Bureau of the Budget coordinating the government programs that collected accounting statistics; his 1945 paper describing his wartime activities was the first recipient of an award from the American Institute of Accountants for the best paper of the year.

Cooper began his academic career with a brief teaching stint, from 1944 to 1946, back at the University of Chicago. In 1945, Cooper married his wife Ruth, a lawyer and human activist, and in 1946 he joined the newly formed Graduate School of Industrial Administration at the Carnegie Institute of Technology (now the Tepper School of Business at Carnegie Mellon University). There, he formed important research collaborations with Abraham Charnes, George Leland Bach, and Herbert A. Simon, and eventually became University Professor. While at CMU, from 1949 to 1950, he also worked again as an assistant to Eric Kohler, who had by this time become Comptroller of the Marshall Plan.In 1969 he left GSIA but stayed at CMU, becoming dean of the new School of Urban and Public Affairs (now the Heinz College) there. As dean, he realized that there would soon be a much greater role in American business management for African-Americans, and worked to increase African-American representation within the school.

In 1975, Harvard University hired Cooper away from CMU to become the Dickinson Professor of Accounting, and in 1980 he moved again, to the University of Texas at Austin, where he became the Foster Parker Professor of Management, Finance and Accounting. He retired in 1993, but continued to be active in research until his death on June 20, 2012.

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, 19 June 2026

Some Notes on Division, and its History (Including Alien Division for Fractions)

The word Divide shares its major root with the word widow. The root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse. A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel. The prefix, di, of divide is a contraction of dis, a two based word meaning apart or away, as in the process of division in which equal parts are separated from each other. Notice that the vi part of vidua is also derived from a two word, and is the same root as in vigesimal (two tens), for things related to twenty. An individual is one who can not be divided.

In a division problem such as 24 / 6 = 4 the number being divided, in this case the 24, is called the dividend and the number that is being used to divide it, the 6, is called the divisor. The four is called the quotient. If the quotient is not a factor of the dividend, then some quantity will remain after division. This quantity is usually called the remainder, although residue sometimes is used. The Treviso Arithmetic uses the word lauanzo for remainder. In Frank Swetz's book, Capitalism and Arithmetic he gives, "The term lauanzo apparently evolved from l'avenzo, meaning a surplus, or in a business context, a profit." Swetz also points out that in the 15th Century the term partition (partire in Latin) was synonymous with the word divisision.

In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,

"It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near Nürnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use."

I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10):

From the fourteenth century on, merchants from the north traveled to Italy, particularly to Venice, to learn the arte de mercadanta, the mercantile art, of the Italians. Sons of German businessman flocked to Venice to study...

Early algorithms for division:
By the middle ages there seem to have been five approaches to the process of division.

The first was called the Galley, galea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term “scratch” has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Recorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. " Addendum (I just found a note that says that the famous Arab mathematician, Al-Khwarizmi, whose book laid the foundation for western introduction to the now-common Arabic numbers, used this method in his writing. The Hindu mathematical calculations were usually done on a dust covered tablet, and they would wipe out numbers instead of scratching through them so that in the final result, only results appeared. The scratch-out method in Europe was simply an adaptation for paper or slates. )

Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda.

The page the image is from has a nice step by step illustration of the process.

In 2005 I acquired a German student "copy book" from 1804 which seems to show the Galley division method and the student's illustration of the ship around the work.

A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from Ray's New Practical Arithmetic published in 1877. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others .

This method of subtraction grew into what is taught frequently as an alternative use to many modern classrooms, and I believe is the standard method at some levels for a program called "Everyday Math". Instead of slowly subtracting one divisor at a time, the use of simple multiples is used to group subtract.

For example, to divide 227 by 8, it is easy to see that 10 x 8 or eighty can be subtracted, so they might take out ten groups of eight repeatedly until the remaining part was too small to divide by 80. So after removing 20 groups of eight, there would be 67 remaining. At this point they might recognize that 8x8 is 64 so by removing 8 more sets of eight would leave only three remaining, so the quotient would be 28 with 3 remaining.
A video of this method is shown here.
(I have been informed that the "correct" term for this method is "partial quotients".)

A third method was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.

divide 92467 by 168 or 4 x 6 x 7

4|92467

6|23116 …. groups of four and 3 units over

7| 3852 ….. groups of 24 (4x6) and 4 foursover

___550 groups of 168 and 2 twenty-fours over

The complete remainder is 2 (24) + 4(4) + 3 = 67

A fourth method is presented in the Liber Abaci, by Fibonacci in 1202. After introducing how to divide by numbers of one digit,
and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A composed fraction might look like image at right. Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right.
The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75.

When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four.

This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of 749 divided by 75 as 9 and 4/5 + 4/25 + 2/75 or 9 74/75.

A fifth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method appears in another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400’s up to the early 20th century.

The sixth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.

In a Gresham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."

Later as I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . (It seems when I just checked that the Met no longer allows that link, will replace ASAP) The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.

I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one
interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.

The Agony and the Obelus, much Ado about Notation

Recently (in 2015) James Tanton posted a short article about problems that are circulating on the internet such as (and this is the one he used) "What is the value of the following expression: 6² ÷ 2(3)+4, and then asked, "Is the answer 10 or is the answer 58?" (my personal choice for historical reasons explained below is 3.6)

I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.

The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols

Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunately, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.

The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write

\underline{2 x + 3} 5

. Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.

Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).

The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking. Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, †. The root also gives rise to the word obelisk for a pointed pillar of stone.
The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659.
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.

Let me make it clear I am not an authority on math history and do not read German, but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog. Instead it seems to be used exclusively for a shorthand in explaining the operations used.

Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.

The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.

Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 6² ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)

So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means

\frac{a}{b c}

. He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean

\frac{a}{b} (c)

So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first." So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that. Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."

So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven. The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets. What was not there was a section on order of operations, or any problems that went beyond " number ÷ number." No long strings of numbers and operations strung together.

Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher

Read the part in Italics again.... multiplication first, then division, without regard to the order. That is not what you are telling your students today (I hope). So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.

Better, right? then turn the page, and find

So there is our old friend the obelus used exactly as I suspect Pell and Rahn had intended (if they intended it to be used as an operator at all), and lower down the solidus in the manner that Stokes suggested, but apparently used in a way the users thought distinguished it from the use of the obelus. And you wonder why your students are confused?

I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem. Anyone who has more information, please share.
I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.

Division of Fractions by the Alien Method (and followup)

I wrote about an experience that happened when I let my kids watch an old science fiction movie in class just before Christmas... The blog, and a followup requested by a teacher who admitted he wasn't really sure why the common "divide and multiply method worked... Here they are as a package...
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The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... ²/₃ x ⁵/₇ = ¹⁰/₂₁. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3dogs = 5" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.

A few days after I wrote that blog I got a response that asked, more or less, "OK, why does the common algorithm work?"
This was my response

I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.
For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... because the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...
The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this situation describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.
We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previous number) and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.
At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.
Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.
What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.
Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write