Thursday, 7 May 2026

On This Day in Math - May 7

  



Scientists study the world as it is,
engineers create the world that never has been

~Theodore Von Karman

The 127th day of the year.  127 is the fourth Mersienne Prime, 27-1.   Édouard Lucas verified 2127-1 as prime in 1876. He is said to have spent 19 years in checking this 39 digit prime by hand. This remains the largest prime number discovered without the aid of a computer. (Lucas also invented the Towers of Hanoi Puzzle, and the game of dots and boxes which he called "La Pipopipette".)

20 + 21 + 22 + 23 + 24 + 25 + 26 = 127.

127 can be expressed as the sum of factorials of the first three odd numbers (1! + 3! + 5!). 

127 is the smallest odd number that can't be written as a prime P + 2ˣ for some integer.  

17=2^2+13, 19=2^4+3,... 125 =2^6+61 , 127 .....  129 = 2^5+97

127 x \(\sqrt{62}\) is almost an integer, 999.998999999...

 And in a rare equivalence, 127 cm is equal to 50 inches. HT Don S. McDonald ‏@McDONewt



EVENTS

1526 The first circumnavigation of the globe took place in 1519. In 1539 Cardano asked for the number of days spent if a ship sailed westward on January 1, 1517, and went three times around the earth, returning on May 7, 1526. See Sanford, History, pp. 214 and 377. *VFR

1660 Isaack B Fubine of Savoy, in The Hague, patents macaroni *TIS (as soon as someone invents cheese, the fun eating will begin)




1747 Johann Sebastian Bach visits King Frederick II of Prussia, the visit resulting in his Musikalische Opfer (Musical offering). See D. R. Hofstadter’s Godel, Escher, Bach, p. 4. [Manson]*VFR

1772 Read before the Royal Society, May 7, The Sieve of Eratosthenes. Being an Account of His Method of Finding All the Prime Numbers, by the Rev. Samuel Horsley, F. R. S.
================================================================
1811  On this day in 1811, Babbage, Herschel and Peacock first met each other to discuss the possible formation of a society whose aim would be to encourage the study of Leibniz's analytical methods in Cambridge. The formal inaugural meeting of the Analytical Society took place very shortly afterwards. *SAU
*SAU



1895 Steiger Gets "Millionaire" Patent:Otto Steiger was issued a patent for his Millionaire calculating machine. For the next 40 years, Switzerland's Hans Egli manufactured 4,700 machines, which weighed 120 pounds each. The Millionaire was notable in its ability to perform direct multiplication, which meant a user could multiply a number by a single digit with a single rotation of the handle.*CHM

In his German Patent of 1892 Steiger describes a machine which uses a mechanical representation of the multiplication table to form partial products, in the same way that a human "calculator" uses a
multiplication table committed to memory. The partial products are then transferred via a "transmitting mechanism" to a "combining and registering mechanism" for display to the operator. The Steiger's machine is to be regarded as a proper multiplication machine in that it solves problems of multiplication directly on the basis of the multiplication table, whereas other types of calculating machines are only adding machines and, as such, carry out multiplication by a continued series of additions. *Georgi Dalakov, History of Computers    


Radio wave coherer, built by Alexander Popov
1895 
 Nearly everyone knows that Guglielmo Marconi was the inventor of radio, but nearly everyone is wrong about this. Marconi received the Nobel Prize in 1909 for the "development" of radio, not the invention – lots of other people were ahead of him here. Nikolai Tesla has many proponents in the United States as the inventor of radio, while in England they give the credit to Oliver Lodge, who in 1894 invented the first detector, called a coherer, which reacted to the presence of radio waves, a key step if one wants to use radio waves to communicate. It is easy to generate radio waves, but detecting them is much more difficult, and Lodge's coherer allowed one to do just that.
But if you ask anyone in Russia who invented radio, they will tell you: Alexander Stepanovich Popov. And they have a good case. On May 7, 1895 (this would be after Lodge but before Tesla and Marconi), Popov demonstrated a radio receiver to the Russian Physical and Chemical Society in St. Petersburg, and he published a paper on his device later that year. In March of 1896, Popov is supposed to have transmitted the wireless message "Heinrich Hertz" between two buildings on the campus in St. Petersburg (Hertz, a German physicist, was the first to predict the possibility of radio waves in 1888). *Linda Hall Library

In 1952the concept of the integrated circuit chip was first presented, at a Symposium on Progress in Quality Electronic Components in Washington DC., by radar scientist Geoffrey W.A. Dummer. His small team of researchers at the Royal Radar Establishment of the British Ministry of Defence, based at Malvern, Worcestershire, was working on the task of improving the reliability of the Royal Air Force's radar equipment.. He believed that it would be possible to fabricate multiple circuit elements on and into a block of silicon half an inch square. In 1956, his initial attempts to build such a circuit failed, and thereafter could get no further support for his idea. Britain lost the commercial lead. A few years later, in America, Jack Kilby of Texas Instruments was awarded a U.S. patent for essentially the same idea.*TIS
Dummer



1954 , construction began on the Mackinac Bridge including the world’s longest suspension bridge to date. It fulfilled the 70 year dream to connect 8 km (5 miles) across the Straits of Mackinac between Michigan’s upper and lower peninsulas. The architect was David B. Steinman. Ceremonial ground breakings were held on this day at the St. Ignace end, and the next day on the opposite shore at Mackinaw City. Components were already being assembled in several states. Caissons were floated into position and sunk to provide the footings. On 1 Nov 1957, the bridge was opened to traffic, the automobile ferry service ended, and travel time was cut from about 2 hours to 10 minutes. It was dedicated 28 Jun 1958.




1963, the United States launched the Telstar 2 communications satellite on behalf of its private owner, AT&T. On its tenth orbit, it transmitted the first transatlantic TV program seen in colour. It orbited with an apogee of 6,700 miles (10,800 km). This superceded AT&T's original Telstar satellite, which had ceased operating in 1962, due to transistor damage caused by radiation from a high-altitude nuclear test. Telstar 2 was built with shielding against such radiation.





BIRTHS

Alexis Clairaut (sometimes Clairault) (7 May 1713; 17 May 1765 at age 51) was a French mathematician who worked to confirm the Newton-Huygens belief that the Earth was flattened at the poles. He was a child prodigy was studying calculus at age 10 and was admitted to the Academy of Sciences at age 18. He was the first person to estimate the mass of Venus to a close value. He also calculated the return date of Halley's comet. In about 1737, Pierre de Maupertuis led an expedition (including Clairaut) to measure a degree along a meridian in Lapland, while Bouguer and La Condamine went to Peru.  The results, even before the Peru expedition had returned, showed that Newton was correct in predicting that the earth was flattened at the poles .(various) A nice brief summary of Clairaut's life and works is here.

Title page of Alexis Clairaut, Théorie de la figure de la terre, tirée des principes de l'hydrostatique, 1743 (Linda Hall Library)







1774 Sir Francis Beaufort (7 May 1774; 17 Dec 1857 at age 83)British naval officer, who devised (1805) a scale of wind force from 0 (calm) to 12 (hurricane) which was based on observation and so required no special instruments. [Chase]*VFR The initial scale of thirteen classes (zero to twelve) did not reference wind speed numbers but related qualitative wind conditions to effects on the sails of a man-of-war, then the main ship of the Royal Navy, from "just sufficient to give steerage" to "that which no canvas sails could withstand".
Although he devised the scale in 1805, it would not be adopted by the Royal Navy until 1830 when was an administrator. The first official use of the scale in a ships log was on December 22, 1831 by Robert Fitzroy on the first day of Darwin's voyage on the Beagle.
In 1829 Beaufort became the British Admiralty Hydrographer of the Navy. He remained at the post for 25 years. Beaufort converted what had been a minor chart repository into the finest surveying and charting institution in the world. Some of the excellent charts the Office produced are still in use today.

During his tenure, he took over the administration of the great astronomical observatories at Greenwich, England, and the Cape of Good Hope, Africa. Beaufort directed some of the major maritime explorations and experiments of that period. For eight years, Beaufort directed the Arctic Council during its search for the explorer, Sir John Franklin, lost in his last polar voyage to search for the legendary Northwest Passage. *Wik





1832 Carl Gottfried Neumann (May 7, 1832 - March 27, 1925) He worked on a wide range of topics in applied mathematics such as mathematical physics, potential theory and electrodynamics. He also made important pure mathematical contributions. He studied the order of connectivity of Riemann surfaces.
During the 1860s Neumann wrote papers on the Dirichlet principle and the 'logarithmic potential', a term he coined. In 1890 Émile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equations.*SAU



1854 Giuseppe Veronese (7 May 1854 – 17 July 1917) invented non-Archimedean geometries which he proposed around 1890. However Peano strongly criticised the notion due to the lack of rigor of Veronese's description and also for the fact that he did not justify his use of infinitesimal and infinite segments. The resulting argument was extremely useful to mathematics since it helped to clarify the notion of the continuum. Any fears that non-Archimedean systems would not be consistent were shown to unnecessary soon after this when Hilbert proved that indeed they were consistent.*SAU




1880 Oskar Perron(7 May 1880 – 22 February 1975) was a German mathematician best known for the Perron paradox:
Suppose the largest natural number is N. Then if N is greater than 1 we have N2 greater than  N contradicting the definition.  *SAU
He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential equations and partial differential equations, including the Perron method to solve the Dirichlet problem for elliptic partial differential equations. He wrote an encyclopedic book on continued fractions Die Lehre von den Kettenbrüchen. He introduced Perron's paradox to illustrate the danger of assuming that the solution of an optimization problem exists. *Wik




1911 Raymond Arthur Lyttleton (7 May 1911; 16 May 1995 at age 83) English mathematician and theoretical astronomer who researched stellar evolution and composition. In 1939, with Fred Hoyle, he demonstrated the large scale existence of interstellar hydrogen, refuting the existing belief of that space was devoid of interstellar gas. Together, in the early 1940's, they applied nuclear physics to explain how energy is generated by stars. In his own mongraph (1953) Lyttleton described stability of rotating liquid masses, which he extended later to explain that the Earth had a liquid core resulting from a phase change associated with a combination of intense pressure and temperature. With Hermann Bondi, in 1959, he proposed the electrostatic theory of the expanding universe. He authored various astronomy books.*TIS





1914  Johannes de Groot (May 7, 1914 – September 11, 1972) was a Dutch mathematician, the leading Dutch topologist for more than two decades following World War II
De Groot published approximately 90 scientific papers. His mathematical research concerned, in general, topology and topological group theory, although he also made contributions to abstract algebra and mathematical analysis.

He wrote several papers on dimension theory (a topic that had also been of interest to Brouwer). His first work on this subject, in his thesis, concerned the compactness degree of a space: this is a number, defined to be −1 for a compact space, and 1 + x if every point in the space has a neighbourhood the boundary of which has compactness degree x. He made an important conjecture, only solved much later in 1982 by Pol and 1988 by Kimura, that the compactness degree was the same as the minimum dimension of a set that could be adjoined to the space to compactify it. Thus, for instance the familiar Euclidean space has compactness degree zero; it is not compact itself, but every point has a neighborhood bounded by a compact sphere. This compactness degree, zero, equals the dimension of the single point that may be added to Euclidean space to form its one-point compactification. A detailed review of de Groot's compactness degree problem and its relation to other definitions of dimension for topological spaces is provided by Koetsier and van Mill.  




1921 Bent Christiansen (7 May, 1921 - 3 Sept, 1996)  From his Obituary: "Bent was a legend in mathematics education in Denmark and the Nordic countries. His impact on the development of the teaching and learning of mathematics in primary and lower secondary education can hardly be over-estimated. He wrote textbooks and books on mathematics education, especially the very influential 'Goals and means in basic mathematics education' ('Mål og midler I den elementære matematikundervisning', 1967). He gave innumerable in-service courses and invited lectures at meetings and conferences. Naturally, he also served on hosts of national committees, including the Danish National Sub-Commission of ICMI (1961-1972). All this earned him a reputation as a charismatic, enthusiastic and extremely energetic mentor for generations of mathematics teachers, teacher trainers and colleagues."




1927 Allen Shields (7 May 1927 New York – 16 September 1989 Ann Arbor, Michigan, USA) worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis and operator theory.
A special issue of The Mathematical Intelligencer, for which he served as editor of the "Years Ago" column, was dedicated to his memory in 1990.
*Wik




1939  Sidney Altman (May 7, 1939 – April 5, 2022) was a Canadian-American[1] molecular biologist, who was the Sterling Professor of Molecular, Cellular, and Developmental Biology and Chemistry at Yale University. In 1989, he shared the Nobel Prize in Chemistry with Thomas R. Cech for their work on the catalytic properties of RNA.
While at Yale, Altman's Nobel Prize work came with the analysis of the catalytic properties of the ribozyme RNase P, a ribonucleoprotein particle consisting of both a structural RNA molecule and one (in prokaryotes) or more (in eukaryotes) proteins. Originally, it was believed that, in the bacterial RNase P complex, the protein subunit was responsible for the catalytic activity of the complex, which is involved in the maturation of tRNAs. During experiments in which the complex was reconstituted in test tubes, Altman and his group discovered that the RNA component, in isolation, was sufficient for the observed catalytic activity of the enzyme, indicating that the RNA itself had catalytic properties, which was the discovery that earned him the Nobel Prize. Although the RNase P complex also exists in eukaryotic organisms, his later work revealed that in those organisms, the protein subunits of the complex are essential to the catalytic activity, in contrast to the bacterial RNase P.
Altman was elected a Fellow of the American Academy of Arts and Sciences in 1988[8] and a member of both the National Academy of Sciences and the American Philosophical Society in 1990.  *Wik




DEATHS

1617 David Fabricius, (March 9, 1564 – May 7, 1617)a Protestant minister, was killed by a parishioner angered upon being accused by him as a thief.   A German astronomer, friend of Tycho Brahe and Kepler, and one of the first to follow Galileo in telescope observation of the skies. He is best known for a naked-eye observation of a star on Aug 3, 1596, subsequently named Omicron Ceti, the first variable star to be discovered, and now known as Mira. Its existence with variable brightness contradicted the Aristotelian dogma that the heavens were both perfect and constant. With his son, Johannes Fabricius, he observed the sun and noted sunspots. For further observations they invented the use of a camera obscura and recorded sun-spot motion indicating the rotation of the Sun. *TIS  [re: invented, The Camera Obscura (Latin for dark room) was a dark box or room with a hole in one end. If the hole was small enough, an inverted image would be seen on the opposite wall. Such a principle was known by thinkers as early as Aristotle (c. 300 BC). It is said that Roger Bacon invented the camera obscura just before the year 1300, but this has never been accepted by scholars; more plausible is the claim that he used one to observe solar eclipses. In fact, the Arabian scholar Hassan ibn Hassan (also known as Ibn al Haitam), in the 10th century, described what can be called a camera obscura in his writings..]






1934 Karl Friedrich Geiser (26 Feb 1843 in Langenthal, Bern, Switzerland, 7 May 1934 in Küsnacht, Zürich, Switzerland) Swiss mathematician who worked in algebraic geometry and minimal surfaces. He organised the first International Mathematical Congress in Zurich.*SAU
In addition to his research results, Geiser's participation in the development of Switzerland's education system is remarkable. He was helped by his relationships (partly due to his family connection with Jakob Steiner) with eminent politicians and mathematicians. Geiser published previously unpublished lecture notes and treatises from Steiner's Nachlass. Geiser and Ferdinand Rudio were two of the main organizers of the International Congress of Mathematicians in 1897 in Zürich.*Wik




1963  Theodore von Karman (May 11, 1881 – May 7, 1963)Hungarian-American aerospace engineer and physicist who was active primarily in the fields of aeronautics and astronautics. He is responsible for many key advances in aerodynamics, notably his work on supersonic and hypersonic airflow characterization.*Wikipedia [another who died very close to his birthday (May 11), someday I must do statistics on this.]  He was director of the Institute for Aerodynamics at the Rheinisch-Westfälische Technische Hochschule (RWTH) in AACHEN, Nordrhein-Westfalen, in 1913-1934. The main lecture theatre complex is named the Kármán Auditorium and there is a photo and a bust of him in the foyer.  He is buried in a vault in the Hollywood Forever Cemetery in Los Angeles, Ca. USA




2007 Emma Markovna Lehmer (née Trotskaia) (November 6, 1906 – May 7, 2007) was a mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather than the more abstract aspects of the theory. At UC Berkeley, she started out in engineering in 1924, but found her niche in mathematics. One of her professors was Derrick N. Lehmer, the number theorist well known for his work on prime number tables and factorizations. While working for him at Berkeley finding pseudosquares, she met her future husband Derrick H. Lehmer. Upon her graduation summa cum laude with a B.A. in Mathematics (1928), Emma married the younger Lehmer. They moved to Brown University, where Emma received her M.Sc., and Derrick his Ph.D., both in 1930. Emma did not obtain a Ph.D. herself. Most universities had nepotism rules which prevented husband and wife from both holding teaching positions, although Emma claimed there were many advantages to not holding a Ph.D.
The Lehmers had two children, Laura (1932) and Donald (1934). Emma did independent mathematical work, including a translation from Russian to English of Pontryagin's book Topological Groups. She worked closely with her husband on many projects; 21 of her 60-some publications were joint work with him. Her publications were mainly in number theory and computation, with emphasis on reciprocity laws, special primes, and congruences. *Wik
Photo by Paul Halmos


*SAU




2017  Wu Wenjun (Chinese: 吴文俊; 12 May 1919 – 7 May 2017), also commonly known as Wu Wen-tsün, was a Chinese mathematician, historian, and writer. He was an academician at the Chinese Academy of Sciences (CAS), best known for Wu class, Wu formula, and Wu's method of characteristic set.
He was also active in the field of the history of Chinese mathematics. He was the chief editor of the ten-volume Grand Series of Chinese Mathematics, covering the time from antiquity to late part of the Qin dynasty.
In 1957, he was elected as an academician of the Chinese Academy of Sciences. In 1986 he was an Invited Speaker of the ICM in Berkeley. In 1990, he was elected as an academician of The World Academy of Sciences (TWAS).

Along with Yuan Longping, he was awarded the State Preeminent Science and Technology Award by President Jiang Zemin in 2000, when this highest scientific and technological prize in China began to be awarded. He also received the TWAS Prize in 1990[3] and the Shaw Prize in 2006. He was the President of the Chinese society of mathematics. He died on May 7, 2017, 5 days before his 98th birthday.




2018 Peter Andreas Grünberg (German pronunciation: [18 May 1939 – 7 April 2018) was a German physicist, and Nobel Prize in Physics laureate for his discovery with Albert Fert of giant magnetoresistance which brought about a breakthrough in gigabyte hard disk drives.
In 1986 he discovered the antiparallel exchange coupling between ferromagnetic layers separated by a thin non-ferromagnetic layer, and in 1988 he discovered the giant magnetoresistive effect (GMR). GMR was simultaneously and independently discovered by Albert Fert from the Université de Paris Sud. It has been used extensively in read heads of modern hard drives. Another application of the GMR effect is non-volatile, magnetic random access memory.

Apart from the Nobel Prize, work also has been rewarded with shared prizes in the APS International Prize for New Materials, the International Union of Pure and Applied Physics Magnetism Award, the Hewlett-Packard Europhysics Prize, the Wolf Prize in Physics and the 2007 Japan Prize. He won the German Future Prize for Technology and Innovation in 1998 and was named European Inventor of the Year in the category "Universities and research institutions" by the European Patent Office and European Commission in 2006.



2025 Jacob Palis Jr. (15 March 1940 – 7 May 2025) was a Brazilian mathematician and academic. Palis's research interests were mainly dynamical systems and differential equations. Some themes are global stability and hyperbolicity, bifurcations, attractors and chaotic systems. He proposed the Palis' conjectures (which form the Palis' program), which influenced the development of the theory of dynamical systems, and also of its applications to other sciences.[6] He was a world leader in chaos theory research. Palis was an influential figure in the development of mathematics in Brazil.
 Palis was born in Uberaba, Minas Gerais. His father was a Syrian immigrant, and his mother was of Lebanese ancestry. The couple had eight children (five men and three women), and Jacob was the youngest. His father was a merchant, owner of a large store, and supported and funded the studies of his children. Palis said that he already enjoyed mathematics in his childhood.

At 16, Palis moved to Rio de Janeiro to study engineering at the University of Brazil – now UFRJ. He was approved in first place in the entrance exam, but was not old enough to be accepted; he then had to take the university's entry exam again a year later, at which again he obtained first place. He completed the course in 1962 with honors and receiving the award for the best student.

In 1964, he moved to the United States. In 1966, he obtained his master's degree in mathematics under the guidance of Stephen Smale at the University of California, Berkeley, and in 1968 his PhD, with the thesis On Morse-Smale Diffeomorphisms, again with Smale as advisor.

In 1968, he returned to Brazil and became a researcher at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil. Beginning in 1973 he held a permanent position as professor at IMPA, where he was director from 1993 until 2003. He was Secretary-General of the Third World Academy of Sciences from 2004 to 2006, and elected its president in 2006[13] and remained on position till December 2012. He was also president of the International Mathematical Union from 1999 to 2002. He was president of the Brazilian Academy of Sciences from 2007 to 2016.[15] Palis advised more than forty PhD students so far from more than ten countries, including Artur Oscar Lopes, Ricardo Mañé, Welington de Melo, Carlos Gustavo Moreira, Enrique Pujals and Marcelo Viana

Palis died at a hospital in Rio de Janeiro, on 7 May 2025, at the age of 85. He had been hospitalized since March.

Palis received numerous medals and decorations. He was a foreign member of several academies of sciences, including the United States National Academy of Sciences, the French Academy of Sciences and German Academy of Sciences Leopoldina. In 2005 Palis received the Legion of Honor.

He was a member of the Norwegian Academy of Science and Letters. In 2010, he was awarded the Balzan Prize for his fundamental contributions in the mathematical theory of dynamical systems that has been the basis for many applications in various scientific disciplines, such as in the study of oscillations. He was also a recipient of the 1988 TWAS 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

Wednesday, 6 May 2026

More on a Geome-Treat with a Calculus Twist

 

I recently re-posted a blog I wrote 12 years ago about a way to find the tangent to the curve of a conic without employing calculus that was one of my favorite math "tricks". In response I got a serious question from Brandon@_nilradical who asked, "anything "similar" for cubics, quartics, etc?". Being busy and wanting to reply I passed off a hasty, "Nothing quite so simple." and went back to mowing. Later I felt guilty about dismissing the question, so I thought I would fess up to what little I have found in playing with the idea "Beyond the Quadratics."
(If you are not familiar with the point substitution approach to finding tangents, take a moment to read the older post, it's pretty brief.)


My first excursion was to try a simple cubic and see if I could figure out how to apply the same polar idea. What could be easier than y=x3 so I picked the point (1,1) and set out exploring. I first had to decide how to replace the three x's in the right side, and decided that I would replace y= x3 with (y+1)/2 =(x)(1)(x+1)/2


Ok, we got a tangent, but it was a tangent quadratic, not a tangent line. So I decided to press on and apply the idea recursively into the new parabola.  The parabola simplified to y=x2 +x - 1, so I began to substitute into that using the point (1,1) again.  This leads to (y+1)/2 = (1)(x) + (x+1)/2-1

I didn't even bother to simply, just entered into the Desmos calculator and ....Eureka!!!



So how could we extract this and explain with a little calculus?  Well, if we begin by saying we want to create a parabola at the point (1,1) with the same slope (3) as the cubic there, we would begin with y=x2 + bx +c  and since the derivative of that, y'=2x+b must equal 3, setting 3=2(1)+b we see that b must be 1 also.  Now we just plug (1,1) into the equation for y=x2 + x +c and we quickly find that indeed, the calculus will give us y=x2 +x - 1 as the parabola tangent to the point (1,1) with the same slope as y=x3

At this point I had no idea what this descending cascade of polar approaches to a tangent would do with something really complicated, but I barged ahead and created something minorly absurd.
So I started typing into the calculator creating as I go along and came up with x3y = y2 +x+x4

Desmos responds with :





Ok, this looks like fun.  The first problem is finding a nice point with integers and a non-zero slope.... and (-1,0) jumps out because it should eliminate some congestion substituting y=0 in some places.

I begin by the same approach of using equal parts of variable and constant wherever possible, and write out (-1)(x)(x+1)/2 (y+0)/2= 0y +(x-1)/2 + (-1)(-1)(x)(x)  and we get a cubic with two infinite discontinuities, but one of the branches slashes through the point we seek



At this point I'm convinced that our descending iteration of polars will proceed to a line tangent at the same point... and I found it interesting that even picking the equation out of my head, the point I chose also had a slope of 3 at that point.

I'm not sure I have any idea how useful these techniques might be for those forging father than my simple experiences in math can anticipate, but if you are one who knows more about this, share what you know.


I posted tongue-in-cheek at the end of the first post that the challenge of proving that it always worked was a homework assignment for calculus students, due by Wednesday.  In response, Thomas Morgan replied:  "I can verify that this will work in general. Multivariable calculus tells us that to compute the tangent vector at a point, one need only compute the partial derivatives at that point. It should be clear from the method description that the new curve intersects the old curve at the point under consideration. Two different curves that intersect at a point are tangent at that point if their tangent vector points in the same direction, so we can scale our tangent if necessary. Using partial derivatives, in addition to the sum rule for derivatives, reduces the effort to computing single-variable derivatives for two cases. In the first case, consider the tangent of c*x^(2n) at the point x=t. Computing the derivative directly yields 2n*c*x^(2n-1). Using your method, we first transform c*x^(2n) to c*x^n*t^n. Its derivative is n*c*x^(n-1)*t^n, which at the point x=t only differs from the first calculation by a factor of 2. We similarly compute the derivative of c*x^(2n+1) at the point x=t. Direct calculation yields (2n+1)*c*x^(2n). Using your method, we first transform c*x^(2n+1) into (c*x^(n+1)*t^n + c*x^n*t^(n+1))/2. Its derivative is ((n+1)*c*x^n*t^n + n*c*x^(n-1)*t^(n+1))/2, which at the point x=t only differs from the first calculation by a factor of 2. Thus we see that the tangent vector to the curve given by your transformation is the same vector as the tangent vector of the original curve, compressed by a factor of 2. This shows that the new curve is tangent to the original curve, and by induction, we are done."  

Thank you for the response Thomas.  

On This Day in Math - May 6

  




*Fermat's Library


The 126th day of the year; nine points around a circle form the vertices of \( \binom{n}{k} = 126 \)  unique quadrilaterals.   That also means that if you draw all the diagonals of the nonagon, you would be using the same 126 sets of four vertices to get 126 intersections. 

126 is the first of four consecutive numbers that are the sum of a cube and a square. 5³+ 1²


In non-leap  years, there are 126 days in which the day of the month is prime.

The prime gap that covers the first century with no primes (from 1671800 to 1671899) has length 126 (from 1671781 to 1671907). 

There are 9 choose 5, or 126 ways for a random selection to pick the five spaces on a tic tac toe board for the "first player" in a random game.  36 of these configurations are a "win" for both players. They have both three x's and three O's in a line, since they don't have an order of play. Over 58% of those games are a win for the "first player". Geometrically, a student could think of each random game as a pentagon selected from nine points spaced around a circle.

125 and 126 are a Ruth Aaron pair of the second kind.  In the first kind prime factors are only counted once, in the second kind they are counted as often as they appear, so 5+5+5 =  2+3+3+7, The original kind were discovered for 714 Ruth's career record, and 715, the number on the day Aaron passed his record (he went on to get more).   







EVENTS

1604 Longomontanus wrote to Kepler criticizing his attacks on Tycho's system using Tycho's data: "These and perhaps all other things that were discovered and worked out by Tycho during his restoration of astronomy for our eternal benefit, you, my dear Kepler, although submerged in shit in the Augean stable of old, do not scruple to equal." More detail at the Renaissance Mathematicus blog



1747 Euler to Goldbach, QED  Euler succeeded in proving Fermat's theorem on sums of two squares in 1747, when he was forty years old. He communicated this in a letter to Goldbach dated 6 May 1747. The proof relies on infinite descent.  Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as
p = x2 + y2
with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Fermat in 1640, but he supplied no proof. *Wik




1775 After being driven out of Concord by an angry mob because of his Tory leanings, the American born Benjamin Thompson, later to become Count Rumford, sends the first invisible ink letter of the American Revolution  . Within a few days of the Battle of Lexington, British headquarters in Boston received a secret ink letter which revealed details of the military plans of the patriot forces in New England. Long suspected to have been from Thompson in his home town of Woburn, Mass, recent chemical and handwriting analysis have conclusively confirmed that it was from him. *American Journal of Police Science




1807 Bessel wrote to Gauss, "I saw with pleasure that you have calculated the orbit of Vesta; also the name chosen by you is splendid, and therefore certainly also pleasant to all your friends because it shows them to which goddess you sacrifice.". Carl Friedrich Gauss: Titan of Science  By Guy Waldo Dunnington, Jeremy Gray, Fritz-Egbert Dohse 

Vesta  is the virgin goddess of the hearth, home, and family in Roman religion. She was rarely depicted in human form, and was more often represented by the fire of her temple in the Forum Romanum. Entry to her temple was permitted only to her priestesses, the Vestal Virgins.
Vesta's most important role was as the guardian of the sacred fire in every household and in the city's public hearth. 



1840, the adhesive postage stamp was first sold in Great Britain. The "penny black" and "twopenny blue" stamps showed the profile of Queen Victoria. *TIS


1889 The Eiffel Tower, 7e, was built in 26 months and opened in Mar 1889 for the Universal Exposition.  it is 320.75 m (1051 ft) high and only weighs 7000 tons – less than the air around it!  The tower was inaugurated on 31 March 1889, and opened on 6 May. I recently read that "Gustav Eiffel included a flat for himself at the top of the Eiffel Tower, and retired there at age 62 to conduct aerodynamic experiments".
Today, after being off limits for years, the apartment is on display for visitors to come and peer into. Much of the furnishings remain the same and there are a couple of rather wan looking mannequins of Eiffel and Edison. For the right type of architectural admirer Eiffel’s secret apartment could inspire as much jealousy today as it did when it was built.
In 2016 several stories below Gustave Eiffel’s private apartment, a second Eiffel Tower apartment opened temporarily in the summer of 2016. Vacation rental company HomeAway transformed an unused conference space on the first floor of the tower into a new pop-op designer apartment as a marketing promotion. Four contest winners got a chance to stay inside the tower in July, 2016. *Atlas Obscura



1896  Samuel Pierpont Langley became the third Secretary of the Smithsonian Institution in 1887. In 1891, he began experiments with large, tandem-winged models powered by small steam and gasoline engines he called aerodromes. After several failures with designs that were too fragile and under-powered to sustain themselves, Langley had his first genuine success on May 6, 1896, with his Aerodrome Number 5. It made the world's first successful flight of an unpiloted, engine-driven, heavier-than-air craft of substantial size. It was launched from a spring-actuated catapult mounted on top of a houseboat on the Potomac River near Quantico, Virginia. Two flights were made on May 6, one of 1,005 m (3,300 ft) and a second of 700 m (2,300 ft), at a speed of approximately 40 kph (25 mph). On both occasions, the Aerodrome Number 5 landed in the water, as planned, because, in order to save weight, it was not equipped with landing gear.

*Wikimedia


1937  The Hindenburg disaster was an airship accident that occurred on May 6, 1937, in Manchester Township, New Jersey, U.S. The LZ 129 Hindenburg (Luftschiff Zeppelin #129; Registration: D-LZ 129) was a German commercial passenger-carrying rigid airship, the lead ship of the Hindenburg class, the longest class of flying machine and the largest airship by envelope volume.[1] It was designed and built by the Zeppelin Company (Luftschiffbau Zeppelin GmbH) and was operated by the German Zeppelin Airline Company (Deutsche Zeppelin-Reederei). It was named after Field Marshal Paul von Hindenburg, who was president of Germany from 1925 until his death in 1934. Filled with hydrogen, it caught fire and was destroyed during its attempt to dock with its mooring mast at Naval Air Station Lakehurst. The accident caused 35 fatalities (13 passengers and 22 crewmen) from the 97 people on board (36 passengers and 61 crewmen), and an additional fatality on the ground.




1950 A famous series begins on this day. Can you guess what it is? The first terms are Nicky Hilton, Michael Wilding, Mike Todd, Eddie Fisher, Richard Burton, Richard Burton, John Warner, Larry Fortensky. Note that one term in the series repeats; that’s perfectly natural. Your are right, this is a Taylor series. More specifically, the Elizabeth Taylor series. These are her husbands (eight by last count (2010)). *VFR [sadly, Ms. Taylor has departed the matrimonial game... at least on this sphere. pb]

1949 British Computer EDSAC Performs First Calculation. The EDSAC performed its first calculation. Maurice Wilkes had assembled the machine -- the first practical stored-program computer -- at Cambridge University (an earlier machine at the University of Manchester was too small for practical purposes). His ideas grew out of the Moore School lectures he had attended three years earlier at the University of Pennsylvania. For programming the EDSAC, Wilkes established a library of short programs called subroutines stored on punched paper tapes. It performed 714 operations per second. *CHM
*Computer History Museum, Mountain View, CA


1954 Roger Bannister defied the general belief that it was impossible to run a mile in less than 4 minutes by running one in 3 minutes 59.4 seconds. *VFR  ..."the announcer at the Oxford University cinder track in England calmly gave the placings in the one mile race, and then started to announce the winning time, beginning with the word “three...” The small crowd erupted in delirious excitement, the rest of the announcement went unheard"*Runner's World
At the 1952 Olympics in Helsinki, Bannister set a British record in the 1500 metres and finished in fourth place. This achievement strengthened his resolve to become the first athlete to finish the mile run in under four minutes. He accomplished this feat on 6 May 1954 at Iffley Road track in Oxford, with Chris Chataway and Chris Brasher providing the pacing.
Bannister went on to become a neurologist and Master of Pembroke College, Oxford, before retiring in 1993. As Master of Pembroke, he was on the governing body of Abingdon School from 1986 to 1993. When asked whether the 4-minute mile was his proudest achievement, he said he felt prouder of his contribution to academic medicine through research into the responses of the nervous system. Bannister was patron of the MSA Trust. He was diagnosed with Parkinson's disease in 2011.  *HT Offer Pade’




1996, Volodymyr Petryshyn,a professor of mathematics at Rutgers University, killed his wife, Ukrainian-American painter Arcadia Olenska-Petryshyn, with a hammer. After making a mistake in a proof he feared that he would be ridiculed by his fellow mathematicians and while under the strain of this, he had a complete mental breakdown. He was found not guilty of murder by reason of insanity. He later published Development of mathematical sciences in the Ukraine in Ukrainian in 2004.   Petryshyn died on March 21, 2020





BIRTHS

1635 Johann Joachim Becher, (1635–1682),the German physician and alchemist who initiated the theory that became the phlogiston theory, born. William Cullen considered Becher as a chemist of first importance and Physica Subterranea as the most considerable of Bechers writings. He reintroduced Paracelsus’ tria prima in the form of three different types of Earth.
  • terra fluida or mercurial Earth giving material the characteristics, fluidity, fineness, fugacity, metallic appearance
  • terra pinguis or fatty Earth giving material the characteristics oily, sulphurous and flammable
  • terra lapidea glassy Earth, giving material the characteristic fusibility
It was the second of these, terra pingus, that  was adopted into phlogisten theory.  For a longer, clearer, and more knowledgeable look at this development read this by The Renaissance Mathematicus. Like George Box's comment on statistical models, "Wrong, but useful".






1667 Abraham De Moivre ( 26 May 1667 – 27 November 1754) born in Vitry-le-Francois, Champagne, France. *VFR [.. a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.
De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci numbers linking the nth power of φ (the so-called "golden ratio" to the nth Fibonacci number.](Wikipedia)




1769 Jean Nicolas Pierre Hachette (May 6, 1769 – January 16, 1834) worked on descriptive geometry, collected work by Monge and edited Monge's Géométrie descriptive which was published in 1799. He also published on a wide range of topics from his own major works on geometry, to works on applied mechanics including the theory of machines. His work on machines includes much in the area of applied mechanics, but he was also interested in applied hydrodynamics and steam engines. In fact he published interesting work on the history of steam engines. *SAU



1792 Martin Ohm (6 May 1792 in Erlangen, Bavaria (now Germany)- 1 April 1872 in Berlin, Prussia, German Empire) was a German mathematician and a younger brother of physicist Georg Ohm. He earned his doctorate in 1811 at Friedrich-Alexander-University, Erlangen-Nuremberg where his advisor was Karl Christian von Langsdorf. Ohm was the first to fully develop the theory of the exponential ab when both a and b are complex numbers in 1823. He is also often credited with introducing the name "golden section" (goldener Schnitt).
Ohm's students included Friedrich August, Friedrich Bachmann, Paul Bachmann, Joseph Brutkowski, Heinrich Eduard Heine, Rudolf Lipschitz, Leo Pochhammer, Friedrich Prym, Wilhelm Wagner, Hermann Waldaestel, Wilhelm Wernicke, Elena Gerz, Valentien Gerz, and Johanna Gerz. *Wik
Martin Ohm made a distinction between writing for mathematicians and writing for students, a distinction that many of his contemporaries, including Hermann Grassmann, did not consider appropriate. His colleagues Steiner and Kummer also ridiculed him for not following Alexander von Humboldt's firm belief in the unity of teaching and research. It is quite difficult to assess the importance of Ohm's mathematical contributions. The first thing to say is that they certainly weren't as important as he himself thought. He had a very high opinion of himself as the following quotation indicates. Niels Abel wrote to Christopher Hansteen, the professor of astronomy at the University of Christiania, while he was on a visit to Berlin in 1826

There is at [August Crelle's] house some kind of meeting where music is mainly discussed, of which unfortunately I do not understand much. I enjoy it all the same since I always meet there some young mathematicians with whom I talk. At Crelle's house, there used to be a meeting of mathematicians, but he had to suspend it because of a certain Martin Ohm with whom nobody could get along due to his terrible arrogance.
*SAU




1872 Willem de Sitter (6 May 1872 – 20 November 1934) Dutch mathematician, astronomer, and cosmologist who developed theoretical models of the universe based on Albert Einstein's general theory of relativity. He worked extensively on the motions of the satellites of Jupiter, determining their masses and orbits from decades of observations. He redetermined the fundamental constants of astronomy and determined the variation of the rotation of the earth. He also performed statistical studies of the distribution and motions of stars, but today he is best known for his contributions to cosmology. His 1917 solution to Albert Einstein's field equations showed that a near-empty universe would expand. Later, he and Einstein found an expanding universe solution without space curvature. *TIS



1906 André Weil  (6 May 1906 – 6 August 1998) was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry. He was a founding member and the de facto early leader of the influential Bourbaki group. The philosopher Simone Weil was his sister.. *Wikipedia




1908 John Frank (Jack) Allen (6 May 1908; 22 Apr 2001 at age 92) was a Canadian physicist who codiscovered the superfluidity of liquid helium near absolute zero temperature. Working at the Royal Society Mond Laboratory in Cambridge, with Don Misener he discovered (1930's) that below 2.17 kelvin temperature, liquid helium could flow through very small capillaries with practically zero viscosity. Independently, P. L. Kapitza in Moscow produced similar results at about the same time. Their two articles were published together in the 8 Jan 1938 issue of the journal Nature. Superfluidity is a visible manifestation resulting from the quantum mechanics of Bose- Einstein condensation. By 1945, research in Moscow delved into the microscopic aspect, which Allen did not pursue.*TIS




1916 Robert Henry Dicke (6 May 1916 St. Louis, Missouri, USA - 4 Mar 1997 at age 80) American physicist who worked in such wide-ranging fields as microwave physics, cosmology, and relativity. As an inspired theorist and a successful experimentalist, his unifying theme was the application of powerful and scrupulously controlled experimental methods to issues that really matter. He also made a number of significant contributions to radar technology and to the field of atomic physics. His visualization of an oscillating universe stimulated the discovery of the cosmic microwave background, the most direct evidence that our universe really did expand from a dense state. A key instrument in measurements of this fossil of the Big Bang is the microwave radiometer he invented. His patents ranged from clothes dryers to lasers. *TIS




DEATHS

1856 William Stirling Hamilton (8 March 1788 in Glasgow, Scotland - 6 May 1856 in Edinburgh, Scotland) Hamilton became professor of logic and metaphysics at the University of Edinburgh, giving his inaugural lecture on 21 November. Hamilton was one of the first in a series of British logicians to create the algebra of logic and introduced the 'quantification of the predicate'. Boole, De Morgan and Venn followed him, but Hamilton helped begin this development and his work, although not of great depth, influenced Boole to produce a much more sophisticated system. Sadly, however, Hamilton claimed that De Morgan was guilty of plagiarism which was a ridiculous suggestion. *SAU




1859  Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 1769 – 6 May 1859) was a German polymath, geographer, naturalist, explorer, and proponent of Romantic philosophy and science. He was the younger brother of the Prussian minister, philosopher, and linguist Wilhelm von Humboldt (1767–1835). Humboldt's quantitative work on botanical geography laid the foundation for the field of biogeography. Humboldt's advocacy of long-term systematic geophysical measurement laid the foundation for modern geomagnetic and meteorological monitoring.
Between 1799 and 1804, Humboldt traveled extensively in the Americas, exploring and describing them for the first time from a modern Western scientific point of view. His description of the journey was written up and published in several volumes over 21 years. Humboldt was one of the first people to propose that the lands bordering the Atlantic Ocean were once joined (South America and Africa in particular).

Humboldt resurrected the use of the word cosmos from the ancient Greek and assigned it to his multivolume treatise, Kosmos, in which he sought to unify diverse branches of scientific knowledge and culture. This important work also motivated a holistic perception of the universe as one interacting entity, which introduced concepts of ecology leading to ideas of environmentalism. In 1800, and again in 1831, he described scientifically, on the basis of observations generated during his travels, local impacts of development causing human-induced climate change.

Humboldt is seen as "the father of ecology" and "the father of environmentalism".  
Isothermal map of the world using Humboldt's data by William Channing Woodbridge


1862 Olry Terquem (16 June 1782 – 6 May 1862) was a French mathematician. He is known for his works in geometry and for founding two scientific journals, one of which was the first journal about the history of mathematics. He was also the pseudonymous author (as Tsarphati) of a sequence of letters advocating radical Reform in Judaism. He was French Jewish.
Terquem translated works concerning artillery, was the author of several textbooks, and became an expert on the history of mathematics. Terquem and Camille-Christophe Gerono were the founding editors of the Nouvelles Annales de Mathématiques in 1842. Terquem also founded another journal in 1855, the Bulletin de Bibliographie, d'Histoire et de Biographie de Mathématiques, which was published as a supplement to the Nouvelles Annales, and he continued editing it until 1861. This was the first journal dedicated to the history of mathematics.

The three marked points that lie on the nine point circle and interior to the triangle were found by Terquem. The point of convergence of the three red lines through the triangle is its orthocenter. He is also known for naming the nine-point circle and fully proving its properties. This is a circle defined from a given triangle that contains nine special points of the triangle. Karl Wilhelm Feuerbach had previously observed that the three feet of the altitudes of a triangle and the three midpoints of its sides all lie on a single circle, but Terquem was the first to prove that this circle also contains the midpoints of the line segments connecting each vertex to the orthocenter of the triangle. He also gave a new proof of Feuerbach's theorem that the nine-point circle is tangent to the incircle and excircles of a triangle.
Terquem's other contributions to mathematics include naming the pedal curve of another curve, and counting the number of perpendicular lines from a point to an algebraic curve as a function of the degree of the curve. He was also the first to observe that the minimum or maximum value of a symmetric function is often obtained by setting all variables equal to each other.
He became an officer of the Legion of Honor in 1852. After he died, his funeral was officiated by Lazare Isidor, the Chief Rabbi of Paris and later of France, and attended by over 12 generals headed by Edmond Le Bœuf.  *Wik


1916 Ágoston Scholtz (27 July 1844 in Kotterbach, Zips district, Austro-Hungary (now Rudnany, Slovakia) Died: 6 May 1916 in Veszprém, Hungary) From 1871 he was a teacher of mathematics and natural philosophy at the Lutheranian Grammar School of Budapest which at that time had been upgraded to become a so called 'chief grammar school', namely one which offered eight years of teaching. This was precisely the school which later was attended by several famous mathematicians such as Johnny von Neumann and Eugene Wigner (or Jenó Pál Wigner as he was called at that time). Scholtz became the school director of the Lutheranian Grammar School in 1875. Unfortunately this excellent school was closed in 1952, and most of its equipment was lost. Due to the initiative and support of its former well-known students, among others Wigner, it was reopened in 1989 after being closed for thirty-seven years. Scholtz's field of research was projective geometry and theory of de
terminants. His results were recorded by Muir in his famous work The history of determinants *SAU


1951 Élie Joseph Cartan  (9 April 1869 – 6 May 1951) worked on continuous groups, Lie algebras, differential equations and geometry. His work achieves a synthesis between these areas. He is one of the most important mathematicians of the first half of the 20C. *SAU  He was one of the earliest "Bourbaki".




1979 Karl Wilhelm Reinmuth (4 April 1892 in Heidelberg – 6 May 1979 in Heidelberg) was a German astronomer and a prolific discoverer of 395 minor planets.  He was the world's foremost asteroid hunter before automated search techniques. His discoveries (1914-1957) of 389 such minor planets include some of the first found outside of the Solar System's asteroid belt. His Ph.D. thesis was 'Photographische Positionsbestimmung von 356 Schultzschen Nebelflecken' (Photographic Location of 356 Schultz's Nebulae, 1916). He had started a few years earlier, in 1912, volunteering his time to assist Maximillian Wolf, director at the Königstuhl Observatory, Heidelberg. He learned how to study photographic plates to find asteroids, from Wolf, the first astronomer to utilize such technique. On 15 Oct 1914, the minor planet (796) Sarita was the first Reinmuth identified. In 1937, he named Hermes, the asteroid which made the closest then known approach to Earth.





1983 Yudell Leo Luke (26 June 1918 – 6 May 1983) was an American mathematician who made significant contributions to the Midwest Research Institute, was awarded the N. T. Veatch award for Distinguished Research and Creative Activity in 1975, and appointed as Curator's Professor at the University of Missouri in 1978, a post he held until his death. Luke published eight books and nearly 100 papers in a wide variety of mathematical areas, ranging from aeronautics to approximation theory. By his own estimation, Luke reviewed over 1800 papers and books throughout his career.*SAU




2009 Chuan-Chih Hsiung (15 Feb 1915 in Shefong, Jiangsi, China - 6 May 2009 in Needham, Massachusetts, USA), also known as Chuan-Chih Hsiung, C C Hsiung, or Xiong Quanzhi, is a notable Chinese-born American differential geometer. He was Professor Emeritus of Mathematics at Lehigh University, Bethleham PA USA.
He is the founder and editor-in-chief of the Journal of Differential Geometry, an influential journal in the domain. During his early age, he focused on projective geometry. His interests were largely extended after his research in Harvard, including two-dimensional Riemannian manifolds with boundary, conformal transformation problems, complex manifold, curvature and characteristic classes, etc. *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