Saturday 30 September 2023

On This Day in Math - September 30

Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
~Lewis Richardson

The 273rd day of the year; 273oK(to the nearest integer)is the freezing point of water, or 0oC

OOOOH wait, 273 = 13*7*3, and 1373 is also prime.. and There are only two sphenic numbers consisting of concatenation of distinct prime numbers, this is the smaller of the two.(sphenic or wedge numbers are products of three distinct primes) *Prime curios

273 is a repdigit in hexdecimal, or base 16 (111) 16^2 + 16 + 1, and in base 20(vigesimal), where it looks like a bad report card (DD)

273 = 47^2 - 44^2 and also 137^2 - 136^2

Prime Curios says that 273^10 - 10^273 is the smallest n for which this expression is prime. There are no more year days that exhibit this relationship.

Prime Curios includes this tasty factoid, the prime factors of 273 = 3 x 7 x 13, and if concatenated in reverse order, 1373, it is prime.

273 is a Palindrome in base 2(100010001). 2^8 + 2^4 + 2^0. Also a Palindrome in base 4 (10101) , 4^4 + 4^2 + 4^0.

273 is the product of two consecutive Fibonacci numbers (13 x 21) and is thus a Golden rectangle Number. Each of these have side ratios that approximate a Golden Rectangle as they grow larger . 21/13=1.615....

There are 273 distinct ways to connect 6 points with 5 straight non-crossing lines. Here they are:

Only 28 are unique up to reflection and rotation,


1717 Colin Maclaurin (1698–1746), age 19, was appointed to the Mathematics Chair at Marischal College, Aberdeen, Scotland. This is the youngest at which anyone has been elected chair (full professor) at a university. (Guinness) In 1725 he was made Professor at Edinburgh University on the recommendation of Newton. *VFR

At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending a thesis on the Power of Gravity, and remained at Glasgow to study divinity until he was 19, when he was elected professor of mathematics in a ten-day competition at Marischal College and University in Aberdeen. This record as the world's youngest professor endured until March 2008, when the record was officially given to Alia Sabur, Alia Sabur.  On 19 February 2008, at 18 years of age (3 days before her 19th birthday), she was appointed to the position of International Professor as Research Liaison with Stony Brook University by the Dept. of Advanced Technology Fusion at Konkuk University in Seoul, South Korea. The position was a temporary, one-year contract which she chose not to renew.

Colin MacLaurin *Wik

1810 The University of Berlin opened. *VFR It is now called The Humboldt University of Berlin and is Berlin's oldest university. It was founded as the University of Berlin (Universität zu Berlin) by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities.*Wik

1846 First use of either for tooth extraction by American dentist William T G Morten.  He would give a public demonstration on October 16 of his use of the anesthesia. 

*Lucio Gelmini@gelminil7

1890 In his desk notes Sir George Biddell Airy writes about his disappointment on finding an error in his calculations of the moon’s motion. “ I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.” *George Biddell Airy and Wilfrid Airy (ed.), Autobiography of Sir George Biddell Airy (1896), 350.

1893 Felix Klein visits Worlds fair in Chicago, then visits many colleges. On this day the New York Mathematical society had a special meeting to honor him. *VFR

1921 William H Schott patented the "hit-and-miss synchronizer for his clocks. The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shott in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK. They were the most accurate pendulum clocks ever commercially produced, and became the highest standard for timekeeping between the 1920s and the 1940s, after which mechanical clocks were superseded by quartz time standards. They were used worldwide in astronomical observatories, naval observatories, in scientific research, and as a primary standard for national time dissemination services. The Shortt was the first clock to be a more accurate timekeeper than the Earth itself; it was used in 1926 to detect tiny seasonal changes (nutation) in the Earth's rotation rate. *Wik


 The first manned rocket-powered flight was made by German auto maker Fritz von Opel. His Sander RAK 1 was a glider powered by sixteen 50 pound thrust rockets. In it, Opel made a successful flight of 75 seconds, covering almost 2 miles near Frankfurt-am-Main, Germany. This was his final foray as a rocket pioneer, having begun by making several test runs (some in secret) of rocket propelled vehicles. He reached a speed of 238 km/h (148 mph) on the Avus track in Berlin on 23 May, 1928, with the RAK 2. Subsequently, riding the RAK 3 on rails, he pushed the world speed record up to 254 km/h (158 mph). The first glider pilot to fly under rocket power, was another German, Friedrich Staner, who flew about 3/4-mile on 11 Jun 1928.*TIS

1938  On this day in 1938, Olga Taussky married Jack Todd. 50 years later she said:-
"My life and my career would have been so different if my Irishman had not come along."
Olga Taussky-Todd was an Austrian born mathematician who worked on algebraic number theory and matrix theory.
John Todd was an Irish-born numerical analyst. 

Olga Taussky photo by Paul Erdos

2010 The ignoble prizes, presented on this date, included an engineering for collecting Whale Snot, and a MANAGEMENT PRIZE: for demonstrating mathematically that organizations would become more efficient if they promoted people at random. See all the 2010 winners here.


1550 Michael Maestlin (30 September 1550, Göppingen – 20 October 1631, Tübingen) was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centerd orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU The first known calculation of the reciprocal of the golden ratio as a decimal of "about 0.6180340" was written in 1597 by Maestlin in a letter to Kepler. He is also remembered for :
Catalogued the Pleiades cluster on 24 December 1579. Eleven stars in the cluster were recorded by Maestlin, and possibly as many as fourteen were observed.
Occultation of Mars by Venus on 13 October 1590, seen by Maestlin at Heidelberg. *Wik

1715 Étienne Bonnot de Condillac (30 Sep 1715; 3 Aug 1780) French philosopher, psychologist, logician, economist, and the leading advocate in France of the ideas of John Locke (1632-1704). In his works La Logique (1780) and La Langue des calculs (1798), Condillac emphasized the importance of language in logical reasoning, stressing the need for a scientifically designed language and for mathematical calculation as its basis. He combined elements of Locke's theory of knowledge with the scientific methodology of Newton; all knowledge springs from the senses and association of ideas. Condillac devoted careful attention to questions surrounding the origins and nature of language, and enhanced contemporary awareness of the importance of the use of language as a scientific instrument.*TIS

1774 Carl Wilhelm Scheele sent a letter to Antoine Lavoisier announcing the discovery of oxygen (O). Unfortunately the letter from the Swedish chemist was never acknowledged and Joseph Priestly published the discovery first. Scheele was trounced in the announcement of other discoveries as well, he identified molybdenum, tungsten, barium, hydrogen, and chlorine before Humphry Davy, among others. Scheele discovered organic acids tartaric, oxalic, uric, lactic, and citric, as well as hydrofluoric, hydrocyanic, and arsenic acids. (Not bad for a chemist you never heard of.) For this reason, Isaac Asimov nicknamed him “hard-luck Scheele” * , *Wik
An interpretation of Scheele from the late 19th or early 20th century as no contemporary portraits of him are known (by xylographer Ida Amanda Maria Falander (1842-1927))

1775 Robert Adrain (30 September 1775 – 10 August 1843) . Although born in Ireland he was one of the first creative mathematicians to work in America. *VFR Adrain was appointed as a master at Princeton Academy and remained there until 1800 when the family moved to York in Pennsylvania. In York Adrain became Principal of York County Academy. When the first mathematics journal, the Mathematical Correspondent, began publishing in 1804 under the editorship of George Baron, Adrain became one of its main contributors. One year later, in 1805, he moved again this time to Reading, also in Pennsylvania, where he was appointed Principal of the Academy.
After arriving in Reading, Adrain continued to publish in the Mathematical Correspondent and, in 1807, he became editor of the journal. One has to understand that publishing a mathematics journal in the United States at this time was not an easy task since there were only two mathematicians capable of work of international standing in the whole country, namely Adrain and Nathaniel Bowditch. Despite these problems, Adrain decided to try publishing his own mathematics journal after he had edited only one volume of the Mathematical Correspondent and, in 1808, he began editing his journal the Analyst or Mathematical Museum.
With so few creative mathematicians in the United States the journal had little chance of success and indeed it ceased publication after only one year. After the journal ceased publication, Adrain was appointed professor of mathematics at Queen's College (now Rutgers University) New Brunswick where he worked from 1809 to 1813. Despite Queen's College trying its best to keep him there, Adrain moved to Columbia College in New York in 1813. He tried to restart his mathematical journal the Analyst in 1814 but only one part appeared. In 1825, while he was still on the staff at Columbia College, Adrain made another attempt at publishing a mathematical journal. Realising that the Analyst had been too high powered for the mathematicians of the United States, he published the Mathematical Diary in 1825. This was a lower level publication which continued under the editorship of James Ryan when Adrain left Columbia College in 1826. *SAU
This is the title page of an American edition of Charles Hutton’s A Course of Mathematics for the Use of Academies as well as Private Tuition. Hutton (1737-1823) was a self taught mathematician and mathematics instructor at the British Royal Military Academy, in Woolwich, England. He wrote this text for his students and included all the mathematics he believed a military cadet of the time should know.

When the United States Military Academy formally opened at West Point in 1801, the study of this book was included for its students. Robert Adrain (1775–1843), a leading American mathematician, revised the book for an American audience in 1812. The copy shown below is the third American edition of 1818.

Eventually, the British version experienced thirteen reprints and its American counterpart went through four reprints. The text was used at West Point until 1823. *MAA Mathematical Treasures

portrait by Charles C. Ingham *Wik

1870 Jean-Baptiste Perrin (30 Sep 1870; 17 Apr 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein's explanation of this phenomenon and thereby confirmed the atomic nature of matter. Using a gamboge emulsion, Perrin was able to determine by a new method, one of the most important physical constants, Avogadro's number (the number of molecules of a substance in so many grams as indicated by the molecular weight, for example, the number of molecules in two grams of hydrogen). The value obtained corresponded, within the limits of error, to that given by the kinetic theory of gases. For this achievement he was honoured with the Nobel Prize for Physics in 1926.*TIS

1882 Hans Wilhelm Geiger  (30 Sep 1882; 24 Sep 1945) was a German physicist who introduced the Geiger counter, the first successful detector of individual alpha particles and other ionizing radiations. After earning his Ph.D. at the University of Erlangen in 1906, he collaborated at the University of Manchester with Ernest Rutherford. He used the first version of his particle counter, and other detectors, in experiments that led to the identification of the alpha particle as the nucleus of the helium atom and to Rutherford's statement (1912) that the nucleus occupies a very small volume in the atom. The Geiger-Müller counter (developed with Walther Müller) had improved durability, performance and sensitivity to detect not only alpha particles but also beta particles (electrons) and ionizing electromagnetic photons. Geiger returned to Germany in 1912 and continued to investigate cosmic rays, artificial radioactivity, and nuclear fission.*TIS
A "two-piece" bench-type Geiger–Müller counter using a cylindrical end-window detector connected to an electronics module with analogue readout

1883 Ernst David Hellinger (1883 - 1950) introduced a new type of integral: the Hellinger integral . Jointly with Hilbert he produced an important theory of forms. *SAU

1894 Dirk Jan Struik (30 Sept 1894 , 21 Oct 2000) Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.
In 1924, funded by a Rockefeller fellowship, Struik traveled to Rome to collaborate with the Italian mathematician Tullio Levi-Civita. It was in Rome that Struik first developed a keen interest in the history of mathematics. In 1925, thanks to an extension of his fellowship, Struik went to Göttingen to work with Richard Courant compiling Felix Klein's lectures on the history of 19th-century mathematics. He also started researching Renaissance mathematics at this time.
Struik was a steadfast Marxist. Having joined the Communist Party of the Netherlands in 1919, he remained a Party member his entire life. When asked, upon the occasion of his 100th birthday, how he managed to pen peer-reviewed journal articles at such an advanced age, Struik replied blithely that he had the "3Ms" a man needs to sustain himself: Marriage (his wife, Saly Ruth Ramler, was not alive when he turned one hundred in 1994), 
Mathematics, and Marxism.
It is therefore not surprising that Dirk suffered persecution during the McCarthyite era. He was accused of being a Soviet spy, a charge he vehemently denied. Invoking the First and Fifth Amendments of the U.S. Constitution, he refused to answer any of the 200 questions put forward to him during the HUAC hearing. He was suspended from teaching for five years (with full salary) by MIT in the 1950s. Struik was re-instated in 1956. He retired from MIT in 1960 as Professor Emeritus of Mathematics.
Aside from purely academic work, Struik also helped found the Journal of Science and Society, a Marxian journal on the history, sociology and development of science.
In 1950 Stuik published his Lectures on Classical Differential Geometry.
Struik's other major works include such classics as A Concise History of Mathematics, Yankee Science in the Making, The Birth of the Communist Manifesto, and A Source Book in Mathematics, 1200-1800, all of which are considered standard textbooks or references.
Struik died October 21, 2000, 21 days after celebrating his 106th birthday. *Wik

1905 Sir Nevill F. Mott (30 Sep 1905; 8 Aug 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

1913 Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish and American mathematician born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg–Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974. *Wik

1916 Richard Kenneth Guy (born September 30, 1916, Nuneaton, Warwickshire -  March 9,  2020 ) is a British mathematician, and Professor Emeritus in the Department of Mathematics at the University of Calgary.
He is best known for co-authorship (with John Conway and Elwyn Berlekamp) of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory, but he has also published over 100 papers and books covering combinatorial game theory, number theory and graph theory.
He is said to have developed the partially tongue-in-cheek "Strong Law of Small Numbers," which says there are not enough small integers available for the many tasks assigned to them — thus explaining many coincidences and patterns found among numerous cultures.
Additionally, around 1959, Guy discovered a unistable polyhedron having only 19 faces; no such construct with fewer faces has yet been found. Guy also discovered the glider in Conway's Game of Life.
Guy is also a notable figure in the field of chess endgame studies. He composed around 200 studies, and was co-inventor of the Guy-Blandford-Roycroft code for classifying studies. He also served as the endgame study editor for the British Chess Magazine from 1948 to 1951.
Guy wrote four papers with Paul Erdős, giving him an Erdős number of 1. He also solved one of Erdős problems.
His son, Michael Guy, is also a computer scientist and mathematician. *Wik

1918 Leslie Fox (30 September 1918 – 1 August 1992) was a British mathematician noted for his contribution to numerical analysis. *Wik


1953 Lewis Fry Richardson, FRS (11 October 1881 - 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work on fractals and a method for solving a system of linear equations known as modified Richardson iteration.
*Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
~Lewis Richardson

1985 Dr. Charles Francis Richter (26 Apr 1900, 30 Sep 1985) was an American seismologist and inventor of the Richter Scale that measures earthquake intensity which he developed with his colleague, Beno Gutenberg, in the early 1930's. The scale assigns numerical ratings to the energy released by earthquakes. Richter used a seismograph (an instrument generally consisting of a constantly unwinding roll of paper, anchored to a fixed place, and a pendulum or magnet suspended with a marking device above the roll) to record actual earth motion during an earthquake. The scale takes into account the instrument's distance from the epicenter. Gutenberg suggested that the scale be logarithmic so, for example, a quake of magnitude 7 would be ten times stronger than a 6.*TIS  The quote "logarithmic plots are a device of the devil" is attributed to Richter.

2014 Martin Lewis Perl (June 24, 1927 – September 30, 2014) was an American physicist who won the Nobel Prize in Physics in 1995 for his discovery of the tau lepton.
He received his Ph.D. from Columbia University in 1955, where his thesis advisor was I.I. Rabi. Perl's thesis described measurements of the nuclear quadrupole moment of sodium, using the atomic beam resonance method that Rabi had won the Nobel Prize in Phyics for in 1944.
Following his Ph.D., Perl spent 8 years at the University of Michigan, where he worked on the physics of strong interactions, using bubble chambers and spark chambers to study the scattering of pions and later neutrons on protons.[1] While at Michigan, Perl and Lawrence W. Jones served as co-advisors to Samuel C. C. Ting, who earned the Nobel Prize in Physics in 1976.
Seeking a simpler interaction mechanism to study, Perl started to consider electron and muon interactions. He had the opportunity to start planning experimental work in this area when he moved in 1963 to the Stanford Linear Accelerator Center (SLAC), then being built in California. He was particularly interested in understanding the muon: why it should interact almost exactly like the electron but be 206.8 times heavier, and why it should decay through the route that it does. Perl chose to look for answers to these questions in experiments on high-energy charged leptons. In addition, he considered the possibility of finding a third generation of lepton through electron-positron collisions. He died after a heart attack at Stanford University Hospital on September 30, 2014 at the age of 87. *Wik

2017 Vladimir Voevodsky (Jun 4, 1966 - Sep 30, 2017) formerly a gifted but restless student who flunked out of college out of boredom before emerging as one of the most brilliant and revolutionary mathematicians of his generation, died on Sept. 30 at his home in Princeton, N.J. He was 51.

Dr. Voevodsky was renowned for founding entirely new fields of mathematics and creating groundbreaking new tools for computers to confirm the accuracy of proofs. In 2002, he was awarded the Fields Medal, which recognizes brilliance and promise in mathematicians under 40.
He was “one of the giants of our time,” Thomas Hales, a mathematician at the University of Pittsburgh, said in an interview. Dr. Voevodsky, he said, transformed every field he touched. In his work using computers, for example, he upended mathematical thinking to such a degree that he changed the meaning of the equals sign.

He added: “His ideas gave a new way for all mathematicians to do what they do, a new foundation. The foundations of math are like a constitutional document that spells out the governing rules all mathematicians agree to play by. He has given us a new constitution.”

Vladimir Voevodsky was born in Moscow. His father, Alexander, directed a laboratory in experimental physics at the Russian Academy of Sciences; his mother, Tatyana Voevodskaya, was a chemistry professor at Moscow University. *obit NYTimes

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 29 September 2023

The Harmony of the Harmonic Mean


Things happen in threes according to the old myth, and in this case it was true. I was doing some research on the early history of a mathematical problem often called the "cistern" problem. You probably know the type; "If one pipe can fill a cistern in 6 hours and another can fill it in four hours, how long would it take both pipes working together." While I was working on that, I got a nice article sent to me on the first proof that the harmonic sequence diverges... and then, I was reading a blog by Dave Marain Math Notationsin which he posed a problem that asked, in its general form, given a square inscribed in a right triangle (with one corner at the right angle of the triangle), what is the length of a side of the square in terms of the legs of the triangle.

So what do all these have in common with each other. dare I say what makes them in "harmony"?.... the answer is Harmony, or at least the mathematical relationship of the harmonic mean.

To the early Greeks, if Nichomachus can be believed, all the means were descriptive of musical relations. Much is often made of the Harmonic Mean in relation to a musical sense, but this may not represent the Greek view. Euclid used the word enarmozein to describe a segment that just fits in a given circle. The word is a form of the word Harmozein which the more competent Greek Scholars tell me means to join or to fit together. Jeff Miller's Web site on the first use of Mathematical terms contains a reference to the very early origin of the harmonic mean, 'A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, 'There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic.' The term harmonic mean was also used by Aristotle. "
My search for the early roots of the cistern problem had taken me back to Heron's Metre'seis around the year fifty of the common era. The problem became a staple in arithmetics and problem books and was used by Alcuin (775) and appears in the Lilavati of Bhaskara (1150). I found the illustration I used on the blog for The First Illustrated Arithmetic a few days ago, from the 1492 arithmetic, Trattato di aritmetica by Filippo Calandri.

The solution to a cistern problem is the harmonic mean of the times taken by each pipe. For example, one problem asks "If one pipe can fill a cistern in three hours, and a second can fill it in five hours, how fast will the two pipes take to fill the cistern if both are opened at once. The solution is given by finding the average rate of fill of the two rates, the harmonic mean of three and five, which is three and three-quarter hours. But as the name "mean" suggest, that's the average rate of the two so working together, they would take one-half the time, one and seven-eighths hours, or about an hour and 53 minutes.

The Harmonic mean is the reciprocal of the mean of the reciprocals of the values, so for values a and b, the harmonic mean is given bywhich for two numbers can be simplified to the more economical
Heron might have been the first recorded example of a cistern problem, but a problem calling on the reader to use the harmonic mean occurs even earlier in the Rhind Mathematical Papyrus, now located in the British Museum, in problem 76. The problem involves making loaves of bread with different qualities, but the solution is still the harmonic mean. (I have learned from David Singmaster's Chronology of Recreational Mathematics that the cistern problem appeared, perhaps 300 years before Heron's use, in China by Chiu Chang Suan Shu (around 150 BC).

The series of terms formed by the reciprocals of the positive integers is a common torment for college students in their first introduction to analysis. The sequencein which each number gets smaller and smaller seems to very slowly approach some upper limit. Even after adding 250,000,000 terms, the sum is still less than twenty, and yet... in the mid 1300's, Nichole d'Oresme showed that it will eventually pass any value you can name. In short, it diverges, slowly, very, very slowly, to infinity. Even when warned, it seems like students want to believe it converges. A well-known anecdote about a teacher trying to get student's to remember that it diverges goes:

"Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”<\blockquote>

By the way, each number in the harmonic series is the harmonic mean of the numbers on each side of it (so 1/2 and 1/4 have a harmonic mean of 1/3), and in fact, of any numbers equally spaced away from it such as 1 and 1/5 also have a harmonic mean of 1/3.

And then, I came across that little problem of a square inscribed in a right triangle. If the two legs are a and b, then the sides of the square will have a length equal to the one-half the harmonic mean of a and b .  More generally, a square inscribe in any triangle with one side along a base will have sides equal to one half the harmonic mean of the base and the altitude to that base.  There are lots of other interesting problems that yield to the use of the harmonic mean, and I mean to write again on that collection.

So I guess things do come in threes, unless I come across another one, but whether it comes in threes or fours, it all seems to work together, in perfect harmony.

Many students who struggle with a different puzzle type problem might want to investigate how it too, relates to the harmonic mean, the one where they ask, If you drive to grandmother's house at 60 miles per hour and drive home at 40 miles per hour, what was your average speed for the round trip?  There are dozens more, so just to get a collection, send your favorite problem related to the harmonic mean, and I'll update as they come along. 

On This Day in Math - September 29


Young man, if I could remember the names of these particles, I would have been a botanist.
~Enrico Fermi

The 272nd day of the year; 272 = 24·17, and is the sum of four consecutive primes (61 + 67 + 71 + 73).

272 is also a Pronic or Heteromecic number, the product of two consecutive factors, 16x17 (which makes it twice a triangular #).

And 272 is a palindrome, and the sum of its digits, 11, is also a palindrome. (can you find the next?) The product of its digits is a perfect number.


1609  Almost exactly a year after the first application for a patent of the telescope, Giambaptista della Porta, the Neapolitan polymath, whose Magia Naturalis of 1589, well known all over Europe, because of a tantalizing hint at what might be accomplished by a combination of a convex and concave lens: ‘With a concave you shall see small things afar off, very clearly; with a convex, things neerer to be greater, but more obscurely: if you know how to fit them both together, you shall see both things afar off, and things neer hand, both greater and clearly.’sends a letter to the founder of the Accademia dei Lincei, Prince Federico Cesi in Rome, with a sketch of an instrument that had just reached him, and he wrote:" It is a small tube of soldered silver, one palm in length, and three finger breadths in diameter, which has a convex glass in the end. There is another tube of the same material four finger breadths long, which enters into the first one, and in the end. It has a concave [glass], which is secured like the first one. If observed with that first tube, faraway things are seen as if they were near, but because the vision does not occur along the perpendicular, they appear obscure and indistinct. When the other concave tube, which produces the opposite effect, is inserted, things will be seen clear and erect and it goes in an out, as in a trombone, so that it adjusts to the eyesight of [particular] observers, which all differ. *Albert Van Helden, Galileo and the telescope; Origins of the Telescope, Royal Netherlands Academy of Arts andSciences, 2010
(I assume that we can safely date the invention of the trombone prior to 1609 also)

1762  John Winthrop writes Benjamin Franklin about an article Franklin had written in 1754:

"Cambridge, N.E. Sept. 29, 1762.


There is an observation relating to electricity in the atmosphere, which seemed new to me, though perhaps it will not to you: However, I will venture to mention it. I have some points on the top of my house, and the wire where it passes within-side the house is furnished with bells, according to your method, to give notice of the passage of the electric fluid.  In summer, these bells generally ring at the approach of a thunder cloud; but cease soon after it begins to rain. In winter, they sometimes, though not very often, ring while it is snowing; but never, that I remember, when it rains. But what was unexpected to me was, that, though the bells had not rung while it was snowing, yet the next day, after it had done snowing, and the weather was cleared up; while the snow was driven about by a high wind at W. or N.W. the bells rung for several hours (though with little intermissions) as briskly as ever I knew them, and I drew considerable sparks from the wire. This phaenomenon I never observed but twice; viz. on the 31st of January, 1760, and the 3d of March, 1762. 

I am, Sir, &c."  

**In September 1753 Franklin wrote Collinson that a year earlier he had “erected an Iron Rod to draw the Lightning down into my House, in order to make some Experiments on it, with two Bells to give Notice when the Rod should be electrified.” That letter was published in Exper. and Obser., 1754 edit., and Winthrop probably read it there. *Founder's Archives

1801 Gauss’s Disquisitiones Arithmeticae published. It is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own.
The book is divided into seven sections, which are :
Section I. Congruent Numbers in General
Section II. Congruences of the First Degree
Section III. Residues of Powers
Section IV. Congruences of the Second Degree
Section V. Forms and Indeterminate Equations of the Second Degree
Section VI. Various Applications of the Preceding Discussions
Section VII. Equations Defining Sections of a Circle.
Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He was also the first mathematician to realize the importance of the property of unique factorization (sometimes called the fundamental theorem of arithmetic), which he states and proves explicitly.
From Section IV onwards, much of the work is original. Section IV itself develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary quadratic forms; and Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible i.e. can be constructed with a compass and unmarked straight edge alone. *Wik

1954 The European Organization for Nuclear Research (CERN) was officially established.The organisation runs the world’s largest particle physics laboratory in Geneva, Switzerland. In September 2011 CERN scientists reported that some particles appeared to be travelling faster than light, although it’s now thought that the experiment was flawed. *

Cern Aerial *wik

In 1988, the space shuttle Discovery blasted off from Cape Canaveral, Fla., marking America's return to manned space flight following the Challenger disaster. *TIS

1994 HotJava ---- Programmers first demonstrated the HotJava prototype to executives at Sun Microsystems Inc. A browser making use of Java technology, HotJava attempted to transfer Sun's new programming platform for use on the World Wide Web. Java is based on the concept of being truly universal, allowing an application written in the language to be used on a computer with any type of operating system or on the web, televisions or telephones.*CHM

HotJava 3.0 under Windows XP. *Wik

2001   On the 100th anniversary of his birth, the  scientist who was known in his life as "The Pope" because he made so few mistakes, was memorialized on the occasion of the centennial of his birth with a stamp...that captured a mental mistake in a photo from 1948. In the upper left of the stamp, the symbols h and e are interchanged.  The equation a= h^2/(e*c)  should read a= e^2 /(h*c).  Whether this was a prop written by someone else, or a mental mistake by his own hand, it remains a tongue in cheek memorial to the great, but fallible, scientist.   Something to think about as your tongue comes out to lick a stamp next time, check the address, we all make mistakes. 


1511 (or 1509) Michael Servetus (/sərˈviːtəs/; Spanish: Miguel Serveto as real name, French: Michel Servet), also known as Miguel Servet, Miguel de Villanueva, Michel Servet, Revés, or Michel de Villeneuve (Tudela, Navarre, 29 September 1511 – 27 October 1553), was a Spanish theologian, physician, cartographer, and Renaissance humanist. He was the first European to correctly describe the function of pulmonary circulation, as discussed in Christianismi Restitutio (1553). He was a polymath versed in many sciences: mathematics, astronomy and meteorology, geography, human anatomy, medicine and pharmacology, as well as jurisprudence, translation, poetry and the scholarly study of the Bible in its original languages.
He is renowned in the history of several of these fields, particularly medicine. He participated in the Protestant Reformation, and later rejected the Trinity doctrine and mainstream Catholic Christology. After being condemned by Catholic authorities in France, he fled to Calvinist Geneva where he was burnt at the stake for heresy by order of the city's governing council.

1561  Adriaan van Roomen (29 Sept 1561 , 4 May 1615) is often known by his Latin name Adrianus Romanus. After studying at the Jesuit College in Cologne, Roomen studied medicine at Louvain. He then spent some time in Italy, particularly with Clavius in Rome in 1585.
Roomen was professor of mathematics and medicine at Louvain from 1586 to 1592, he then went to Würzburg where again he was professor of medicine. He was also "Mathematician to the Chapter" in Würzburg. From 1603 to 1610 he lived frequently in both Louvain and Würzburg. He was ordained a priest in 1604. After 1610 he tutored mathematics in Poland.
One of Roomen's most impressive results was finding π to 16 decimal places. He did this in 1593 using 230 sided polygons. Roomen's interest in π was almost certainly as a result of his friendship with Ludolph van Ceulen.
Roomen proposed a problem which involved solving an equation of degree 45. The problem was solved by Viète who realised that there was an underlying trigonometric relation. After this a friendship grew up between the two men. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596.
Roomen worked on trigonometry and the calculation of chords in a circle. In 1596 Rheticus's trigonometric tables Opus palatinum de triangulis were published, many years after Rheticus died. Roomen was critical of the accuracy of the tables and wrote to Clavius at the Collegio Romano in Rome pointing out that, to calculate tangent and secant tables correctly to ten decimal places, it was necessary to work to 20 decimal places for small values of sine, see [2]. In 1600 Roomen visited Prague where he met Kepler and told him of his worries about the methods employed in Rheticus's trigonometric tables. *SAU

Figure 2: Page in Ideae mathematicae,1593, Adriaan van Roomen.

1803   Jacques Charles-François Sturm (29 Sep 1803; 18 Dec 1855) French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations. .While a tutor of the de Broglie family in Paris (1823-24), Sturm met many of the leading French scientists and mathematicians. In 1826, with Swiss engineer Daniel Colladon, he made the first accurate determination of the velocity of sound in water. A year later wrote a prizewinning essay on compressible fluids. Since the time of René Descartes, a problem had existed of finding the number of solutions of a given second-order differential equation within a given range of the variable. Sturm provided a complete solution to the problem with his theorem which first appeared in Mémoire sur la résolution des équations numériques (1829; “Treatise on Numerical Equations”). Those principles have been applied in the development of quantum mechanics, as in the solution of the Schrödinger equation and its boundary values. *TIS  Sturm is also remembered for the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.*SAU

1812  Gustav Adolph Göpel (29 Sept 1812, 7 June 1847) Göpel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms. He wrote on Steiner's synthetic geometry and an important work, Theoriae transcendentium Abelianarum primi ordinis adumbratio levis, published after his death, continued the work of Jacobi on elliptic functions. This work was published in Crelle's Journal in 1847. *SAU

1895 Harold Hotelling​, 29 September 1895 - 26 December 1973   He originally studied journalism at the University of Washington, earning a degree in it in 1919, but eventually turned to mathematics, gaining a PhD in Mathematics from Princeton in 1924 for a dissertation dealing with topology. However, he became interested in statistics that used higher-level math, leading him to go to England in 1929 to study with Fisher.
Although Hotelling first went to Stanford University in 1931, he not many years afterwards became a Professor of Economics at Columbia University, where he helped create Columbia's Stat Dept. In 1946, Hotelling was recruited by Gertrude Cox​ to form a new Stat Dept at the University of North Carolina at Chapel Hill. He became Professor and Chairman of the Dept of Mathematical Statistics, Professor of Economics, and Associate Director of the Institute of Statistics at UNC-CH. (When Hotelling and his wife first arrived in Chapel Hill they instituted the "Hotelling Tea", where they opened their home to students and faculty for tea time once a month.)
Dr. Hotelling's major contributions to statistical theory were in multivariate analysis, with probably his most important paper his famous 1931 paper "The Generalization of Student's Ratio", now known as Hotelling's T^2, which involves a generalization of Student's t-test for multivariate data. In 1953, Hotelling published a 30-plus-page paper on the distribution of the correlation coefficient, following up on the work of Florence Nightingale David in 1938. *David Bee

Probability density function, Horelling's T square  

1901 Enrico Fermi (29 Sep 1901; 28 Nov 1954) Italian-American physicist who was awarded the Nobel Prize for physics in 1938 as one of the chief architects of the nuclear age. He was the last of the double-threat physicists: a genius at creating both esoteric theories and elegant experiments. In 1933, he developed the theory of beta decay, postulating that the newly-discovered neutron decaying to a proton emits an electron and a particle he called a neutrino. Developing theory to explain this decay later resulted in finding the weak interaction force. He developed the mathematical statistics required to clarify a large class of subatomic phenomena, discovered neutron-induced radioactivity, and directed the first controlled chain reaction involving nuclear fission. *TIS

1925 Paul Beattie MacCready (29 Sep 1925; 28 Aug 2007) was an American engineer who invented not only the first human-powered flying machines, but also the first solar-powered aircraft to make sustained flights. On 23 Aug 1977, the pedal-powered aircraft, the Gossamer Condor successfully flew a 1.15 mile figure-8 course to demonstrate sustained, maneuverable manpowered flight, for which he won the £50,000 ($95,000) Kremer Prize. MacCready designed the Condor with Dr. Peter Lissamen. Its frame was made of thin aluminum tubes, covered with mylar plastic supported with stainless steel wire. In 1979, the Gossamer Albatross won the second Kremer Prize for making a flight across the English Channel.*TIS

Gossamer Condor

1931   James Watson Cronin (29 Sep 1931, ) American particle physicist, who shared (with Val Logsdon Fitch) the 1980 Nobel Prize for Physics for "the discovery of violations of fundamental symmetry principles in the decay of neutral K-mesons." Their experiment proved that a reaction run in reverse does not follow the path of the original reaction, which implied that time has an effect on subatomic-particle interactions. Thus the experiment demonstrated a break in particle-antiparticle symmetry for certain reactions of subatomic particles.*TIS

1935 Hillel (Harry) Fürstenberg (September 29, 1935, ..)) is an American-Israeli mathematician, a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. He gained attention at an early stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on a compact hyperbolic Riemann surfaces in the early 1970s. In 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédi's theorem. The Fürstenberg boundary and Fürstenberg compactification of a locally symmetric space are named after him. *Wik


1939 Samuel Dickstein (May 12, 1851 – September 29, 1939) was a Polish mathematician of Jewish origin. He was one of the founders of the Jewish party "Zjednoczenie" (Unification), which advocated the assimilation of Polish Jews.
He was born in Warsaw and was killed there by a German bomb at the beginning of World War II. All the members of his family were killed during the Holocaust.
Dickstein wrote many mathematical books and founded the journal Wiadomości Mathematyczne (Mathematical News), now published by the Polish Mathematical Society. He was a bridge between the times of Cauchy and Poincaré and those of the Lwów School of Mathematics. He was also thanked by Alexander Macfarlane for contributing to the Bibliography of Quaternions (1904) published by the Quaternion Society.
He was also one of the personalities, who contributed to the foundation of the Warsaw Public Library in 1907.*Wik

1941 Friedrich Engel (26 Dec 1861, 29 Sept 1941)Engel was taught by Klein who recognized that he was the right man to assist Lie. At Klein's suggestion Engel went to work with Lie in Christiania (now Oslo) from 1884 until 1885. In 1885 Engel's Habilitation thesis was accepted by Leipzig and he became a lecturer there. The year after Engel returned to Leipzig from Christiania, Lie was appointed to succeed Klein and the collaboration of Lie and Engel continued.
In 1889 Engel was promoted to assistant professor and, ten years later he was promoted to associate professor. In 1904 he accepted the chair of mathematics at Greifswald when his friend Eduard Study resigned the chair. Engel's final post was the chair of mathematics at Giessen which he accepted in 1913 and he remained there for the rest of his life. In 1931 he retired from the university but continued to work in Giessen.
The collaboration between Engel and Lie led to Theorie der Transformationsgruppen a work on three volumes published between 1888 and 1893. This work was, "... prepared by S Lie with the cooperation of F Engel... " 
In many ways it was Engel who put Lie's ideas into a coherent form and made them widely accessible. From 1922 to 1937 Engel published Lie's collected works in six volumes and prepared a seventh (which in fact was not published until 1960). Engel's efforts in producing Lie's collected works are described as, "... an exceptional service to mathematics in particular, and scholarship in general. Lie's peculiar nature made it necessary for his works to be elucidated by one who knew them intimately and thus Engel's 'Annotations' completed in scope with the text itself. "
Engel also edited Hermann Grassmann's complete works and really only after this was published did Grassmann get the fame which his work deserved. Engel collaborated with Stäckel in studying the history of non-euclidean geometry. He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics. *SAU

1955 L(ouis) L(eon) Thurstone (29 May 1887, 29 Sep 1955)  was an American psychologist who improved psychometrics, the measurement of mental functions, and developed statistical techniques for multiple-factor analysis of performance on psychological tests. In high school, he published a letter in Scientific American on a problem of diversion of water from Niagara Falls; and invented a method of trisecting an angle. At university, Thurstone studied engineering. He designed a patented motion picture projector, later demonstrated in the laboratory of Thomas Edison, with whom Thurstone worked briefly as an assistant. When he began teaching engineering, Thurstone became interested in the learning process and pursued a doctorate in psychology. *TIS

2003 Ovide Arino (24 April 1947 - 29 September 2003) was a mathematician working on delay differential equations. His field of application was population dynamics. He was a quite prolific writer, publishing over 150 articles in his lifetime. He also was very active in terms of student supervision, having supervised about 60 theses in total in about 20 years. Also, he organized or coorganized many scientific events. But, most of all, he was an extremely kind human being, interested in finding the good in everyone he met. *Euromedbiomath

2010 Georges Charpak (1 August 1924 – 29 September 2010) was a French physicist who was awarded the Nobel Prize in Physics in 1992 "for his invention and development of particle detectors, in particular the multiwire proportional chamber". This was the last time a single person was awarded the physics prize. *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

Thursday 28 September 2023

More Geometric Solutions to Quadratics

 A few years back I wrote a paper with the tongue-in-cheek title, Twenty Ways to Solve a Quadratic with reference to the song, "Fifty Ways to Leave Your Lover". It included a variety (actually about 20) of approaches, including some graphic approaches, and some history notes on each method.

Recently while looking into the work of Karl George Christian von Staudt (the 147th anniversary of his death is coming up June 1) I found two more graphic methods I had not previously known. I also found an earlier use of one that I had written about in the article above, plus a very early use that I omitted.
Later I changed the name to "Solving Quadratic Equations By analytic and graphic methods; Including several methods you may never have seen." and posted it at
I will begin with two very early examples and conclude with the von Staudt example.

One of the earliest graphic examples of a solution has to be from Euclid's Elements, in book 2 proposition 11. Euclid's description of the task is to cut a given straight line into two parts so that the rectangle formed by the whole and one of the parts is equal to the square on the second part. If we call the parts of the line b and x, then what we seek is x^2 =(x+b)b or x^2 = bx+b^2.

The construction in the Elements is pretty brief, finding a midpoint and a couple of compass constructions.
To see the image, and Euclid's solution I leave you to the always wonderful web page on the Elements by Professor David Joyce.

When I wrote the paper on solving quadratics, I credited a method (#13 in the list) this way:
13. Real roots by Lill circle. One of the most unusual graphic methods I have ever seen comes from a more general
method of solving algebraic equations first proposed, to my knowledge, by M.E. Lill, in Resolution graphique des équations numériques de tous les degrés..., Nouv. Ann. Math. Ser. 2 6 (1867) 359--362. Lill was supposedly an Artillery Captain, but his method was included in Calcul graphique et nomographie by a more famous French engineer, Maurice d’Ocagne, who called it the “Lill Circle”.
It turns out that before Lill, a young Scottish mathematician named Thomas Carlyle was credited with using the same circle to solve quadratics (To be fair, Lill extended his method to all polynomials, and even complex roots).
In Sir John Leslie's (who died in 1832) "Elements of Geometry" he writes this method "...was suggested to me by Thomas Carlyle, an ingenious young mathematician, and formally my pupil."

Carlyle skipped right to the diameter of the circle in question. Given a quadratic, x2+bx + c=0, plot points at (0,1) and (-b,c) and construct a diameter. The circle with this diameter will intersect the x-axis at the real solutions (if any exist) of the quadratic.
The circle for y=x2 +5x - 6 is shown with the actual parabolic function in red.
Students might be challenged to explain why one end of the diameter will always lie on the function.

In 1908 a little book of less than 100 pages titled Graphic Algebra, written by Arthur Schultze, was published by Macmillan & Co. Schultze was a high school math dept. head at the New York High School of Commerce, and an associate Professor at NYU.
The book is free online in several formats.

On page 47 I found a graphic method of solving quadratic equations I had never seen before. The process uses a standard graph of xy=1. [One of the common approaches when calculators and computers did not exist was to alter an equation to the solution of two equations such as a familiar conic and a straight line.]

I will illustrate his approach with the example he uses in his book. To solve the quadratic equation x2 + 2x - 8 =0 He first makes the simple step of dividing all terms by x to get \( x+2 - \frac{8}{x} = 0 ( x\neq 0)\)

Now by substitution of y= 1/x (or xy=1) we get x+2-8y=0. So where both of these equations are true, must be a solution to the original equation. Simply picking a couple of convenient points to plot the line x+2-8y = 0 he determines that when y= 0, x = -2; and when y= 1, x = 6. So we graph the equation xy=1 and then plot the points (-2,0) and ((6,1) and hope for an intersection.

The x-coordinates of the two intersections (red) give us the solutions x={-4; 2}.

Schultze's little book also uses the method from my paper (15. Using the graph of y = x^2 and y = -bx – c to find real roots.) as a graphic solution using a simple conic (y=x2)

And for me, the treat of the day was Karl George Christian von Staudt's graphic method of solving quadratics because it is so different from all the others I've ever learned. Staudt was a student at Gottingen and worked with Gauss, who was at the time the director of the observatory, becoming a very good mathematical astronomer in his own right. He began as a high school teacher as well. His book Geometrie der Lage (1847) was a landmark in projective geometry. It was the first work to completely free projective geometry from any metrical basis. In 1857 von Staudt contributed a route to number through geometry called the Algebra of throws, and he made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently.
So here is the solution method used by von Staudt. The challenge to teacher (and students) is to see if you can figure out WHY it works (I found this very difficult):
Using the equation x2 - 5x + 6 = 0 we begin by constructing a one unit circle centered at (0,1). Then we construct the points \( \frac {c}{-b} , 0)\) and \( \frac{4}{-b},2)\) For b= -5 and c= 6 we get the points \( \frac {6}{5} , 0)\) and \( \frac{4}{5},2)\)
Construct the straight line through these two points (in red in the image).
The line intersects the circle in the points I, and J, and it is not essential to know their coordinates, but J is (1,1)
Now use (2,0) to project each of these points onto the x-axis. The result is the solutions to the equation. In this case x= {2, 3}

I remember reading about the poet Omar Khayyam who developed a number of geometric solutions to cubics.  He had a nice (tricky, for me ) way to solve equations like x^3 + a^2 x = b.  One example like this was in the work of Cardano after he learned/stole Tartaglia's method for cubics, x^3 + 6x = 20.  A little mental work might find the solution since it is an integer, but long before (about 400 years)  Cardano, Omar had a graphic method somewhat based on the Greek idea of completing the square, you might call it completing the cube.
Here is how it worked.  He broke the problem down into the graph of a circle and a parabola.  The center of the circle was at (b/2a^2,0) and the circle passed through (0,0).  Then he set up a parabola with x^2 = ay.  Working out how he came up with that is a pretty big challenge, but fun when you get it.  
For the Cardano equation x^3 + 6x = 20, 6 is a^2, and b=20, so our circle has a center at (20/(2*6), 0), or just (10/3,0).

x^2 = a y can be simp;ified to y = x^2/a, or in this case, y=x^2/(sqrt(6)).  graphing both on geogebra classic we get:

The intersection at x=2 is our real solution for this problem.

The next thing I thought to try was the famous equation that Rafael Bombelli used to get the whole complex number plain in gear.  It was similar, but with a quirk, x^3 = 15x + 4.  The 15x was on the wrong side.  To use Omar K's method above, the 15 would have to be moved tand become negative, but the poet didn't traffic in no crazy negatives.  I didn't have an example of how he had done this kind of cubic, but a moment of inspiration led be to think of what might well have bee his approach.  

Ir we divide all terms by x, we get x^2 = 15 + 4/x.  Since both sides are equal, we can let each of them equal y and so, y= x^2, and y= 15+ 4/x

So I gave them a run on geogebra.
I thought it was interesting that it had solutions at 4 and -4, but I realized it made sense (of course it does, It's math.) when x = +4, 15 + 4/x = 16 and when x = -4, 15 + 4/x = 14.

I found a paper that listed 19 different types of cubics that the Great Poet solved and the method.  I will play with some of these later.


The First Illustrated Arithmetic, and Common Long Division


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 . 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.
Here are some additional notes from my web notes on division that pertain to the long division algorithm and five early methods that were used.

..... 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 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 recent Greasham 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." (I assume he means with the quotient on top.)

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 (which surprised me as it was used almost exclusively in English and American textbooks, and even then seldom beyond elementary school), 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.