Pat'sBlog

Wednesday, 27 May 2026

On This Day in Math - May 27

The mathematics are distinguished by

a particular privilege, that is,

in the course of ages, they may always advance

and can never recede.

~Edward Gibbon, Decline and Fall of the Roman Empire

The 147th day of the year; if you iterate the process of summing the cubes of the digits of a number starting with 147, you eventually start repeating 153. This seems to be true for all multiples of three.

from Das Ambigramm, 147 = 4+5+6..... + 16 + 17 = 18+19+...+ 23+24.

147 is the sum of two Fibonacci numbers, F(12) + F(4 )= 144 + 3 = 147

And 147 = 14^2 - 7^2

If there are no fouls, the maximum score on a snooker break is 147, also known as a maximum or a "one-four-seven". It's achieved by potting (sinking) all 15 reds, then 15 blacks for a total of 120 points, followed by all six colors for an additional 27 points. A 147 is a highly significant achievement in snooker, comparable to a nine-dart finish in darts or a hole-in-one in golf.

And Derek Orr@Derektionary pointed out that "147 is the smallest number formed by a column of numbers on a phone button pad"

147 in binary has an equal number of zeros and ones.

The binary form of 147 (10010011) contains all the two-digit binary numbers (00, 01, 10 and 11).

Strike Fighter Squadron 147 (VFA-147), also known as the "Argonauts," is a United States Navy strike fighter squadron based at MCAS Iwaukuni, Japan.

More math facts for every day of the year here.

EVENTS

669 BC "If the Sun at its rising is like a crescent and wears a crown like the Moon: the king wll capture his enemy's land; evil will leave the land, and (the land) will experience good . . . " Refers to a solar eclipse of 27 May 669 BC. BY Rasil the older, Babylonian scribe to the king. *NSEC

1638 In a letter to Fr Marin Mersenne, Descartes claimed to have a general rule to find number n with a sum of its factors S(n) given only the ratio of n:S(n) = p/q. He showed that n:S(n) = 4/9 is solved for n= 360 . Fermat responded to Mersenne that 2016 has the same property.. (for students, S(6) would = 1+2+3+6 = 12)
1n 1557 Robert Recorde had observed that the aliquot parts of the number 120, add up to 240. Eventually these multiply perfect numbers would become labled \(P_3\) (since the sum of all its divisors, including the number itself, would sum to three times the original number). Mersenne would write to Descartes in 1631, asking if a second such \(P_3\) could be found. After a seven year wait, Descartes responded in Sept. of 1637 with \( P_3 = 673 = 2^5*3*7\). Mersenne would respond that Fermat had a method of finding many such numbers, and had also found 673. The third \(P_3 =523776 = 2^9 * 3 * 11 * 31\) was presented in a letter to Mersenne by Andre Jumeau, the Prior of St. Croix.. In his letter he challenged Descartes to find the fourth.
In his reply on this date, Descartes said that Fermat's rule would provide no other solutions than 120 and 673 . He then proceeds to give the fourth, \( P_3 = 1476304896 = 2^{13} * 3 * 11 * 43 * 127 \) .

Soon after Descartes gave six \(P_4\) :

\(P_4\)(1) = 30240; \(P_4\)(2) =32760 ; \(P_4\)(3) =23569920; \(P_4\)(4) = 142990848; \(P_4\)(5)> = 66433720320; \(P_4\)(6) =403031236608 .
(*History of the theory of numbers By Leonard Eugene Dickson)

1641 Descartes writes to Fr Mersenne again, which was not unusual. The letter wasn't really about math, but about changing his mind about some old disagreements with other philosophers. But then the story got interesting. After Mersenne’s death in 1648, the letter became the property of the French mathematician Gilles de Roberval. When he died in 1675, the French Academy of the Sciences watched over the document for more than a century, until it was stolen by count Guglielmo Libri (1803-1869), a notorious kleptomaniac.

An American collector, Charles Roberts (1846-1902), purchased the letter at an auction in the UK. After his death, he bequeathed his collection to his fellow Quakers at Haverford College.

The previously unknown letter was found by Erik-Jan Bos, a Dutch Historian, through Google. “I regularly browse online. A month ago, I was on one of my little forays when I stumbled upon something I hadn’t seen before.” The document Bos found was a summary of autographs (handwritten, signed texts) that mentioned the letter. The collection the summary referred to is the property of a Quaker-run college in Haverford, Pennsylvania. “They didn’t know this letter had never been published before,” Bos said. The newly discovered letter is only the third by Descartes found in the last 25 years.

When the college learned the letter had been stolen it decided to return it to it former owners. It has since transferred the letter to the French Institute, of which the Academy of the Sciences is a part. *Guardian, *Haverford College

1762 Benjamin Franklin writes to Sir John Pringle, who would become president of the Royal Society in 1772 and physician to King George III in 1774 with a map first naming the "Gulph Stream."

Boston customs officials observed a two-weeks’ difference in the arrival times of ships sailing east to west from England to New York versus England to Rhode Island. He consulted a cousin, Nantucket mariner Timothy Folger, about the problem. Folger was certain that the Gulf Stream was the culprit, for Rhode Island captains were aware of the current through their whaling activities, whereas those of the English packet boats were not. Franklin asked Folger to add the location and dimensions of this current to an available chart so that he could communicate the information to the English sea captains.
Published in England circa 1768, the map was mostly ignored by the stubborn English navigators. Though few copies of this English version seem to have survived (Library of Congress has one), Franklin also had the chart printed in France around 1785, and he published it again with his article “Sundry Maritime Observations” in the Transactions of the American Philosophical Society in 1786. However, it took a long time before the British followed Franklin’s advice on how to avoid fighting this current.

*princeton.edu

1832 In a letter to Legendre, Jacobi stated that the solutions to x²-ay²=1 can be expressed in terms of the sine and cosine of

1849 On this day in 1849, Pafnuty Chebyshev defended his doctoral number theory dissertation The Theory of Congruences at St Petersburg University. This work received a prize from the Academy of Sciences. *MacTutor

Pafnuty Lvovich Chebyshev was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory.

1919 Astronomical party arrives at São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is a Portuguese-speaking island nation in the Gulf of Guinea, off the western equatorial coast of Central Africa. Príncipe was the site where astronomical observations of the total solar eclipse of 29 May 1919 confirmed Einstein's prediction of the curvature of light. The expedition was sponsored by the Royal Society and led by Sir Arthur Stanley Eddington.

Eddington sent one team to Sobral in Brazil, and went himself went to the African island of Príncipe. Stars in the Hyades cluster were behind the sun during the eclipse, and were appeared to shift from their true positions by 1.75 arcseconds. This gravitational deflection of light by the sun's mass provided the first experimental verification of Albert Einstein's theory of general relativity.

1937 Golden Gate bridge opened.*VFR In 1937, the Golden Gate Bridge, San Francisco was first opened to the public as a Pedestrian Day. By 6 am, 18,000 people were waiting for the toll gates to open. Many crossed in unique ways, hoping to be prize-winners as the first to establish a record, whether by walking backwards or on stilts, tap-dancing, roller-skating or playing instruments. It was a sprinter, Donald Bryan, from San Francisco Junior College, who became the first person to cross the entire span. At 10 am, Chief engineer Joseph Strauss gave no speech, but instead read a poem he had written for the event. By the end of the day, about 200,000 people had joined the celebration. The bridge was ceremonially opened to traffic the next day.*TIS

2020 Today’s Doodle, illustrated by Chile-based guest artist Pablo Luebert, celebrates the 95th birthday of a luminary ambassador of the southern night sky: Chilean astrophysicist, author, and professor Adelina Gutiérrez Alonso. Light-years ahead of her time, she was the first Chilean to earn a doctorate in astrophysics, a pioneer not only in her field, but also for women scientists around the world.

Born in the Chilean capital of Santiago on this day in 1925, Carmen Adelina Gutiérrez Alonso was determined from a young age to become a science researcher and teacher. Her scientific career formally took off in 1949, when she joined the faculty at the University of Chile, home of the historic National Astronomical Observatory. In her early years, Adelina crunched data from distant stars, including that collected by her colleague Hugo Moreno León; the two eventually married and formed a fruitful partnership that resulted in a wealth of scientific publications.

But for Adelina, the sky wasn’t the limit. To further her exploration into the mysteries of the cosmos, she moved to the United States in the late 1950s. She graduated from the University of Indiana in 1964 with her unprecedented doctorate in astrophysics, and upon her return home, she helped to establish and lead the country’s first Bachelor of Astronomy program at her alma mater, the University of Chile.

In honor of her stellar scientific contributions, Adelina Gutiérrez Alonso became the first woman and astronomer inducted into the Chilean Academy of Sciences in 1967.

*Google

2021 Ray From Nowhere

On May 27, 2021, an ultrahigh-energy cosmic ray (UHECR) hit Earth’s atmosphere over Utah, sparking showers of subatomic secondary particles that rained down on ground-based detectors. The event measured an estimated 244 exa-electron volts (EeV) in energy, meaning this cosmic ray packed a wallop akin to a well-pitched baseball. Dubbed “Amaterasu” (the goddess of the sun in Japanese mythology) by its discoverers, this single UHECR was the most energetic particle seen on Earth in three decades.

Why this is interesting: Attempts to reconstruct Amaterasu’s path to Earth traced back to the Local Void, a barren expanse of intergalactic space bordering the Milky Way that lacks stars, galaxies and most anything else that could be the ray’s obvious astrophysical source. So where did it really come from? Researchers have no shortage of ideas. But for now no one can even say for sure whether this UHECR was a lightweight proton or something much heavier, like the nucleus of an iron atom—and the distinction matters for plotting Amaterasu’s precise cosmic trajectory, and thus its mysterious origins.

What the experts say: Perhaps Amaterasu’s source “just happens to be a galaxy where a star went fairly close to its supermassive black hole,” says Glennys Farrar, professor of physics at New York University. “I think that’s the most plausible explanation.” *SciAm Today in Science

Amaterasu, born Amaterasu Omikami, is a Shinto deity in Japanese mythology whose name means ''The Great Divinity That Illuminates Heaven.'' Known as the Sun Goddess and ruler of the heavens, Amaterasu is also called Ohirume no Muchi no Kami, which means ''The Great Sun of the Kami.'' Kami are the gods of Shintoism, a Japanese indigenous religion that worships sacred spirits in nature and was the national religion of Japan until the end of World War II. Shinto translates as ''The Way of the Gods'' and comprises ancient beliefs that have survived unchanged for a millennium. Amaterasu Omikami is the Japanese Sun Goddess and has been worshiped for thousands of years. According to Japanese mythology, Amaterasu is an ancestor of the Imperial Family and the daughter of the deities Izanagi and Izanami. Amaterasu represents the transcendent, otherworldly spirit of the universe known as Kunitokotachi.

BIRTHS

1332 Ibn Khaldūn or Ibn Khaldoun (full name, Arabic: أبو زيد عبد الرحمن بن محمد بن خلدون الحضرمي‎, Abū Zayd ‘Abdu r-Raḥmān bin Muḥammad bin Khaldūn Al-Ḥaḍrami, May 27, 1332 AD/732 AH – March 19, 1406 AD/808 AH) was a Muslim historiographer and historian who is often viewed as one of the fathers of modern historiography,sociology and economics.
He is best known for his Muqaddimah (known as Prolegomenon in English), which was discovered, evaluated and fully appreciated first by 19th century European scholarship, although it has also had considerable influence on 17th-century Ottoman historians like Ḥajjī Khalīfa and Mustafa Naima who relied on his theories to analyze the growth and decline of the Ottoman Empire. Later in the 19th century, Western scholars recognized him as one of the greatest philosophers to come out of the Muslim world. *Wik

1660 Francis Hauksbee the elder (baptized on 27 May 1660 in Colchester–buried in St Dunstan's-in-the-West, London on 29 April 1713.), also known as Francis Hawksbee, was an 18th-century English scientist best known for his work on electricity and electrostatic repulsion.
Initially apprenticed in 1678 to his elder brother as a draper, Hauksbee became Isaac Newton’s lab assistant. In 1703 he was appointed curator, instrument maker and experimentalist of the Royal Society by Newton, who had recently become president of the society and wished to resurrect the Royal Society’s weekly demonstrations.
Until 1705, most of these experiments were air pump experiments of a mundane nature, but Hauksbee then turned to investigating the luminosity of mercury which was known to emit a glow under barometric vacuum conditions.
By 1705, Hauksbee had discovered that if he placed a small amount of mercury in the glass of his modified version of Otto von Guericke's generator, evacuated the air from it to create a mild vacuum and rubbed the ball in order to build up a charge, a glow was visible if he placed his hand on the outside of the ball. This glow was bright enough to read by. It seemed to be similar to St. Elmo's Fire. This effect later became the basis of the gas-discharge lamp, which led to neon lighting and mercury vapor lamps. In 1706 he produced an 'Influence machine' to generate this effect. He was elected a Fellow of the Royal Society the same year.

Hauksbee continued to experiment with electricity, making numerous observations and developing machines to generate and demonstrate various electrical phenomena. In 1709 he published Physico-Mechanical Experiments on Various Subjects which summarized much of his scientific work.
In 1708, Hauksbee independently discovered Charles' law of gases, which states that, for a given mass of gas at a constant pressure, the volume of the gas is proportional to its temperature.
The Royal Society Hauksbee Awards, awarded in 2010, were given by the Royal Society to the “unsung heroes of science, technology, engineering and mathematics.” *Wik

1862 John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU His 1903 book, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups, popularized the ideas of Sophus Lie among British mathematicians.

He was elected a Fellow of the Royal Society in 1905, and served as President of the London Mathematical Society from 1918 to 1920. *Wik & *Renaissance Mathematicus

1967 Sir John Douglas Cockcroft (27 May 1897, 18 Sep 1967) British physicist, who shared (with Ernest T.S. Walton of Ireland) the 1951 Nobel Prize for Physics for pioneering the use of particle accelerators to study the atomic nucleus. Together, in 1929, they built an accelerator, the Cockcroft-Walton generator, that generated large numbers of particles at lower energies - the first atom-smasher. In 1932, they used it to disintegrate lithium atoms by bombarding them with protons, the first artificial nuclear reaction not utilizing radioactive substances. They conducted further research on the splitting of other atoms and established the importance of accelerators as a tool for nuclear research. Their accelerator design became one of the most useful in the world's laboratories. *TIS He was the first Master of Churchill College and is buried at the Parish of the Ascension Burial Ground in Cambridge, together with his wife Elizabeth and son John, known as Timothy, who had died at the age of two in 1929.*Wik

1907 Herbert Karl Johannes Seifert (May 27, 1907, Bernstadt – October 1, 1996, Heidelberg) was a German mathematician known for his work in topology. Seifert did other important work related to knot invariants. In 1934 he published results, using surfaces today called Seifert surfaces, which he used to calculate homological knot invariants. Another topic which Seifert worked on was the homeomorphism problem for 3-dimensional closed manifolds. *SAU

a Seifert surface is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface.

*Wik

1909 William Webster Hansen (May 27, 1909 – May 23, 1949) was an American physicist and professor. He was one of the founders of the technology of microwave electronics.

He entered Stanford University at the age of 16, earning his B.A. in 1929 and his Ph.D. in 1933.

Hansen went on to become interested in the problem of accelerating electrons for X-ray experiments, using oscillating fields, rather than large static voltages. At the University of California, Berkeley, Ernest Lawrence and his assistant David H. Sloan, had worked on an accelerator driven by a resonant coil. Hansen proposed replacing the coil with a cavity resonator. In 1937, brothers Russel H. Varian and Sigurd F. Varian came to Stanford to work on the foundations of what was to become radar. Hansen exploited some of the Varian's work to develop the klystron and during the years 1937 to 1940, along with collaborators such as John R. Woodyard, founded the field of microwave electronics. In 1941, he moved his team to the Sperry Gyroscope Company where they spent the war years employing their expertise in radar applications and in other problems.

Returning to Stanford in 1945 as a full professor, he embarked on the construction of a series of linear accelerators based on klystron technology and of GeV performance. Along with the Varian brothers and Edward Ginzton, he co-founded Varian Associates in 1948. Sadly, he was never to see the completion of the klystron project. He died at age 39 in Palo Alto, California of berylliosis and fibrosis of the lungs, caused by inhaling the beryllium used in his research. In 1947, the Hansen Experimental Physics Laboratory (HEPL) was founded as a facility at Stanford University. The facility is designed to promote interdisciplinary enterprises across different branches of science and was named in his honor.

1925 Carmen Adelina Gutiérrez Alonso (aka Adelina Gutiérrez, May 27, 1925 – April 11, 2015) was a Chilean scientist, academic and professor of astrophysics. She was the first Chilean to obtain a doctoral degree in astrophysics and the first woman to become a member of the Chilean Academy of Sciences.

Gutiérrez began working as a science teacher at the Liceo Dario Salas and the Faculty of Physical and Mathematical Sciences (FCFM) of the University of Chile. From June 1, 1949, Gutiérrez worked at National Astronomical Observatory of Chile. At that observatory, her work was initially restricted to analyzing astronomical data obtained by other scientists. While working there, Gutiérrez developed an interest in the photoelectric photometry of austral stars, a subject which she addressed in numerous publications. During the time that she was working at the National Astronomical Observatory, Gutiérrez also became a full faculty member in Faculty of Physical and Mathematical Sciences of the University of Chile.

In the late 1950s, Gutiérrez traveled to the United States to study for a PhD in astrophysics, which has obtained in June 1964, becoming the first Chilean to obtain such a degree. In 1965, after having returned to Chile, Gutiérrez, Hugo Moreno León and Claudio Anguita founded a bachelor's degree course in astronomy at the University of Chile. Gutiérrez was responsible for overseeing the course. In 1976, Gutiérrez also founded a master's degree course in astronomy at the University of Chile.

In 1967, Gutiérrez began working with Hugo Moreno León in the newly opened Cerro Tololo Observatory. That same year she was named a full member of the Chilean Academy of Sciences Institute. She was the first woman and the first astronomer to join that select group of scientists.

1954 Lawrence Maxwell Krauss (born May 27, 1954) is a Canadian-American theoretical physicist and cosmologist who taught at Arizona State University (ASU), , Yale University, and Case Western Reserve University. He founded ASU's Origins Project in 2008 to investigate fundamental questions about the universe and served as the project's director. He was among the first physicists to propose the enigmatic dark energy that makes up most of the mass and energy in the universe. His area of study also includes relating elementary particles to the early universe, general relativity, and neutrino astrophysics. Krauss became the inaugural director of the Origins Project at Arizona State University in Aug 2008. This is a transdisciplinary initiative that nurtures research. Its mission is also to explore fundamental questions, ranging broadly from from the origins of the universe to life; and broaden public understanding of science issues. He has written a number of science books for the layman, including Fear of Physics (1993) and Quantum Man: Richard Feynman's Life in Science (2011). He is also active in popularizing science in print, radio and TV media.

1959 Donna Theo Strickland CC FRS FRSC HonFInstP (born 27 May 1959, ) is a Canadian optical physicist and pioneer in the field of pulsed lasers. She was awarded the Nobel Prize in Physics in 2018, together with Gérard Mourou, for the practical implementation of chirped pulse amplification. She is a professor at the University of Waterloo in Ontario, Canada.

She served as fellow, vice president, and president of Optica (formerly OSA), and is currently chair of its Presidential Advisory Committee. In 2018, she was listed as one of BBC's 100 Women.

From 1988 to 1991, Strickland was a research associate at the National Research Council of Canada, where she worked with Paul Corkum in the Ultrafast Phenomena Section, which had the distinction at that time of having produced the most powerful short-pulse laser in the world. She worked in the laser division of Lawrence Livermore National Laboratory from 1991 to 1992 and joined the technical staff of Princeton University's Advanced Technology Center for Photonics and Opto-electronic Materials in 1992. She joined the University of Waterloo in 1997 as an assistant professor. She became the first full-time female professor in physics at the University of Waterloo. Strickland is currently a professor, leading an ultrafast laser group that develops high-intensity laser systems for nonlinear optics investigations. She has described herself as a "laser jock" *Wik

DEATHS

1781 Giovanni Battista Beccaria FRS (3 October 1716 – 27 May 1781) was an Italian physicist. A fellow of the Royal Society, he published several papers on electrical subjects in the Phil. Trans.

Hespread knowledge of Benjamin Franklin's discoveries with electricity, which he extended with his own research. He designed an electrical thermometer and investigated the relative powers of parallel plate condensers (capacitors). He formed explanations for meteorological and geophysical phenomena in terms of “natural electricity.” With his students, he experimentally probed the atmosphere with metal poles, kites and rockets. He published his work in five books.

*Linda Hall Org

1896 Aleksandr Grigorievich Stoletov (August 10, 1839 – May 27, 1896) was a Russian physicist, founder of electrical engineering, and professor in Moscow University. He was the brother of general Nikolai Stoletov. By the end of the 20th century his disciples had headed the chairs of Physics in five out of seven major universities in Russia.
His major contributions include pioneer work in the field of ferromagnetism and discovery of the laws and principles of the outer photoelectric effect.*Wik

1928 Arthur Moritz Schönflies (April 17, 1853 – May 27, 1928) worked first on geometry and kinematics but became best known for his work on set theory and crystallography. He classified the 230 space groups in 1891 He studied under Kummer and Weierstrass, and was influenced by Felix Klein.
The Schoenflies problem is to prove that an (n − 1)-sphere in Euclidean n-space bounds a topological ball, however embedded. This question is much more subtle than initially appears. *Wik *SAU

1960 Milton B. Porter Professor at Univ of Texas, he was the dissertation adviser for Goldie Horton, the first woman to get a PhD in Mathematics at Univ of Texas. Eighteen years later he married her. He died in Austin Texas.

1962 FELIX ADALBERT BEHREND (23 April 1911 in Charlottenburg, Berlin, Germany -27 May 1962 in Richmond, Victoria, Australia) Felix Behrend's sympathies within pure mathematics were wide, and his creativeness ranged over theory of numbers, algebraic equations, topology, and foundations of analysis. A problem that caught his fancy early and that still occupied him shortly before his death was that of finite models in Euclidean 3-space of the real projective plane. He remained productive for much of the two years of his final illness, and left many unfinished notes in which his work on foundations of analysis is continued. (From his obituary by B H Neumann)

1964 Colin Brian Haselgrove (26 September 1926 – 27 May 1964) In 1958 Haselgrove published his most famous number theory result in A disproof of a conjecture of Pólya. The conjecture of Pólya claims that for every x greater than 1 there are at least as many numbers less than or equal to x having an odd number of prime factors as there are numbers with an even number of prime factors. R S Lehman and W G Spohn had verified the conjecture for all numbers x up to 800,000 but Haselgrove found a counterexample using methods based on those developed by Ingham with the help of computations carried out on the EDSAC 1 computer at Cambridge. He also verified the calculations using Manchester University's Mark I computer before publishing the results. In the same paper Haselgrove announced that he had also disproved a number theory conjecture of Turán. *SAU

1988 Ernst August Friedrich Ruska ( 25 December 1906 – 27 May 1988)[1] was a German physicist who won the Nobel Prize in Physics in 1986 for his work in electron optics, including the design of the first electron microscope. *Wik

For “his fundamental work in electron optics and for the design of the first electron microscope” he was awarded a share of the Nobel Prize for Physics in 1986 (with Heinrich Rohrer and Gerd Binnig). In 1928, found that a magnetic coil could act as a lens to focus an electron beam. Adding a second lens he produced the first primitive (x17 power) electron microscope. By 1933, his refinements increased the magnification to x7000, exceeding what was possible with visible light. The first commercial model was marketed in 1939. Since then, electron microscopes rapidly found applications in biology, medicine and many other areas of science.*TIS

2012 Friedrich Ernst Peter Hirzebruch (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in the Germany of the postwar period.
Amongst many other honours, Hirzebruch was awarded a Wolf Prize in Mathematics in 1988 and a Lobachevsky Medal in 1989. The government of Japan awarded him the Order of the Sacred Treasure in 1996. He also won an Einstein Medal in 1999, and received the Cantor medal in 2004.*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

Tuesday, 26 May 2026

An Unusual Periodic Table

Lemniscate The word lemniscate comes from the Greek word lemniskus for ribbon. The mathematical curve, a sort of figure eight, does look somewhat like the bow for a package made from a twisted ribbon [see figure]. The word is beginning to disappear from textbooks, and is completely missing in my high school edition of the American Heritage Dictionary. The only closely related term I could find was lemniscus, a term for a nerve bundle in the brain. No picture was available, but it may be this also looks like a ribbon.

The Mathematical curve [formulas below]is related to the rectangular hyperbola through the following relationship. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.

I recently saw a picture of a chemical periodic table in the shape of a lemniscate created by William Crookes in 1888. The picture is on page 107 of The Ingredients: A Guided Tour of the Elements by Philip Ball.

On This Day in Math - May 26

You cannot feed the hungry on statistics.

~Heinrich Heine

The 146th day of the year; 146 is 222 in base eight. *What's So Special About This Number
Jim Wilder@wilderlab pointed out that the sum of the divisors of 146; 1+2+73+146 also equals 222.

Finding value of 222 (or other numbers) in base n is nice introduction to polynomials, and (IMHO) leads students to understand polynomials (and base 10) much better.

The decimal expansion of 1/293 has a period of 146 digits.

The concatenation of 146 and 137, 146137 = 317 x 461, both prime. Notice that the same six digits in the product are in the factors, a vampire number. This is the second smallest vampire number with prime factors. (the digits need not be in order, 1260 = 21 × 60 is also a Vampire number.

146 is a number n, for which n²+1 is prime. Goldbach conjectured that any number in this sequence could be written as the sum of two other numbers in the sequence. For 146, one such solution is 146 = 20 + 126 *OEIS

The absolute difference between any two digits of 146 is prime. *Prime Curios, For how many three digit numbers is this true?

Another nice palindrome from Das Ambigram, 146 = 2x5x7+3+3+7x5x2

If you roll two pairs of standard fair dice, the number of ways that both pair can turn out with equal face value showing is 146 out of 1296. The numerator for getting any of the numbers 2 through 12, is an interesting sequence, 1 + 4 + 9 + 16 + 25 + 36 + 25 + 16 + 9 + 4 + 1.... Guess you could say all fair and SQUARE.

146 is Roman Numerals uses all the symbols below 500, CXLVI once each.

EVENTS

1676 Antonie van Leeuwenhoek applied his hobby of making microscopes from his own handmade lenses to observe some water running off a roof during a heavy rainstorm. He finds that it contains, in his words, "very little animalcules." The life he has found in the runoff water is not present in pure rainwater. This was a fundamental discovery, for it showed that the bacteria and one-celled animals did not fall from the sky. When a ball of molten glass is inflated like a balloon, a small droplet of the hot fluid collects at the very bottom the bubble. Leeuwenhoek used these droplets as microscope lenses to view the animalcules. Despite their crude nature, those early lenses enabled Leeuwenhoek to describe an amazing world of microscopic life. * TIS Compound microscopes (that is, microscopes using more than one lens) had been invented around 1595, nearly forty years before Leeuwenhoek was born. Several of Leeuwenhoek's predecessors and contemporaries, notably Robert Hooke in England and Jan Swammerdam in the Netherlands, had built compound microscopes and were making important discoveries with them. These were much more similar to the microscopes in use today. Thus, although Leeuwenhoek is sometimes called "the inventor of the microscope," he was no such thing.

1796 Gauss writes to his counselor, Zimmerman, who had apparently encouraged Gauss to publish the results of his studies on construction of the 17-gon, and the quadratic reciprocity law. Guass wrote that he was prepared to undertake the project, but preferred to write it in German before doing so in Latin where he feared he would be subject to criticism "from another side."

"Since I have an Euler and a Lagrange as predecessors I shall have to marshall great diligence for the composition itself."

*Uta Merzbach, An Early Version of Guass' Disquisitiones Arithmeticae, Mathematical Perspectives, 1981

1896 The Dow Jones Industrial Average was created by Charles Dow and Edward Jones, New York financial reporters. Originally consisted of 11 stocks. They published The Customers Appreciation Letter, which would become the Wall Street Journal. The first index published was on July 3, 1884 *Kane, Famous First Facts

1901 Giuseppe Peano terminated his services to the Royal Military Academy in Turin where he had taught for ﬁfteen years. The trouble was with his teaching. Earlier he was a very good teacher and the author of several excellent texts, but as his work in mathematical logic matured he devoted too much time to what the students called “the symbols.” [H. C. Kennedy, Peano,p. 100] *VFR

1906 German airship designer August von Parseval succeeded launching his new airship at Berlin Tegel military field. In contrast to his rival Zepellin, Parseval’s airships were non-rigid or semi-rigid airships. @SciHiBlog

An airship is any powered, steerable aircraft that it is inflated with a gas that is lighter than air.

“Airship” and “dirigible” are synonyms; a dirigible is any lighter-than-air craft that is powered and steerable, as opposed to free floating like a balloon. The word “dirigible” is often associated with rigid airships but the term does not come from the word “rigid” but from the French verb diriger (“to steer”). Dirigibles include rigid airships (like the Hindenburg), semi-rigid airships (like the Zeppelin NT), and blimps (like the Goodyear blimp).*Airships net

1930 Name for newly discovered planet Pluto announced by United Press. The name had been the suggestion of an English 11 year old girl to her grandfather, a former librarian at the Bodleian Library in Oxford. (see March 14, 1930). I think the fact that PL abbreviation for Pluto (and Percevil Lowell) influenced the folks at the Flagstaff observatory.

1969, the Apollo 10 astronauts returned to Earth after a successful eight-day dress rehearsal for the first manned moon landing. Apollo 10 mission, launched 18 May, was a complete staging of the Apollo 11 mission without actually landing on the Moon. The mission was the second to orbit the Moon and the first to travel to the Moon with the entire Apollo spacecraft configuration. Astronauts Thomas Stafford and Eugene Cernan descended inside the Lunar Module to within 14 kilometers of the lunar surface (achieving the closest approach to the Moon before Apollo 11 landed two months later). Apollo 10 splashed down at 12:52 pm on 26 May, less than 4 miles (6.4 km) from the target point and the recovery ship.

2002 The minor planet 28242 was named after the Mongolian Mathematician Minggatu ( Sharabiin Myangat) as 28242 Mingantu. He was an astronomer, mathematician, and topographic scientist at the Qing court. His courtesy name was Jing An. He was the first person in China who calculated infinite series and obtained more than 10 formulae. In the 1730s, he first established and used what was later to be known as Catalan numbers. The Jesuit missionaries' influence, particularly Pierre Jartoux, can be seen by many traces of European mathematics in his works. *Wik

Ming Antu's geometrical model for trigonometric infinite series.

2021 A renowned astrophysicist and investigator into one of science’s great unsolved mysteries has become the first woman to be appointed as Astronomer Royal for Scotland.
Professor Catherine Heymans, a world-leading expert on the physics of the so-called dark universe, has been awarded the prestigious title, which dates back almost 200 years.
Heymans was recommended to the Queen for the role by an international panel, convened by the Royal Society of Edinburgh.

BIRTHS

1623 Sir William Petty FRS (26 May 1623 – 16 December 1687) was an English economist, scientist and philosopher. He first became prominent serving Oliver Cromwell and Commonwealth in Ireland. He developed efficient methods to survey the land that was to be confiscated and given to Cromwell's soldiers. He also managed to remain prominent under King Charles II and King James II, as did many others who had served Cromwell.
He was Member of the Parliament of England briefly and was also a scientist, inventor, and entrepreneur, and was a charter member of the Royal Society. It is for his theories on economics and his methods of political arithmetic that he is best remembered, however, and to him is attributed the philosophy of 'laissez-faire' in relation to government activity. He was knighted in 1661. He was the great-grandfather of Prime Minister William Petty Fitzmaurice, 2nd Earl of Shelburne and 1st Marquess of Lansdowne.
Petty was a founder member of The Royal Society. He was born and buried in Romsey, and was a friend of Samuel Pepys.
He is best known for economic history and statistic writings, pre-Adam Smith. Of particular interest were Petty's forays into statistical analysis. Petty's work in political arithmetic, along with the work of John Graunt, laid the foundation for modern census techniques. Moreover, this work in statistical analysis, when further expanded by writers like Josiah Child documented some of the first expositions of modern insurance. Vernon Louis Parrington notes him as an early expositor of the labour theory of value as discussed in Treatise of Taxes in 1692.
Petty was knighted in 1661 by Charles II and returned to Ireland in 1666, where he remained for most of the next twenty years. *Wik

1667 Abraham De Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) French mathematician who was a pioneer in the development of analytic trigonometry and in the theory of probability. He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities. He died in poverty, and correctly predicted the day of his own death. He found that he was sleeping 15 minutes longer each night and from this the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. *TIS
Born in Vitry-le-Fran¸cois, France. Being a Protestant, he emigrated to England following the Edict of Nantes in 1685 where he eked out a living as a tutor of mathematics. He became thoroughly Anglicized and pronounced his name “Mowve-re.” *VFR
In Miscellanea Analytica (1730) appears Stirling's formula (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre is also remembered for his formula for (cos x + i sin x)ⁿ which took trigonometry into analysis.

1750 William Morgan, FRS (26 May 1750– 4 May 1833) was a Welsh physician, physicist and statistician, who is considered the father of modern actuarial science. He is also credited with being the first to record the "invisible light" produced when a current is passed through a partly evacuated glass tube: "the first x-ray tube".

He won the Copley Medal in 1789, for his two papers on the values of Reversions and Survivorships, printed in the last two volumes of the Philosophical Transactions of the Royal Society, in the field of actuarial science:

"On the Probabilities of Survivorships Between Two Persons of Any Given Ages, and the Method of Determining the Values of Reversions Depending on those Survivorships", 1788–1794

"On the Method of Determining, from the Real Probabilities of Life, the Value of a Contingent Reversion in Which Three Lives are Involved in the Survivorship". Philosophical Transactions of the Royal Society of London, vol. 79 (1789) pp. 40–54

He was elected a Fellow of the Society, in May of the following year.

Advised by Joseph Priestley, a family friend, he developed an interest in scientific experimentation and is credited with being the first to record the "invisible light" produced when a current is passed through a partly evacuated glass tube: "the first x-ray tube"*Wik

1826 Richard Christopher Carrington (26 May 1826 – 27 November 1875) English astronomer who was the first to map the motions of sunspots and thus discover from them that the Sun rotates faster at the equator than near the poles (equatorial acceleration). He observed that the sunspots were not attached to any solid object, and also discovered the movement of sunspot zones toward the Sun's equator as the solar cycle progresses. On 1 Sep 1859, Carrington was the first to record the observation of a solar flare. *TIS

View of Richard Carrington’s Observatory at Redhill, Surrey, title-page vignette of his A catalogue of 3735 circumpolar stars observed at Redhill, 1857 (Linda Hall Library)

Richard Carrington’s diagram of sunspots, Sep. 1, 1859, with two solar flares marked at A and B, originally published in Monthly Notices of the Royal Astronomical Society, vol. 20, 1859 (welt.de)

1837 Washington Augustus Roebling (May 26, 1837 – July 21, 1926) U.S. civil engineer under whose direction the Brooklyn Bridge, New York City, was completed in 1883. The bridge was designed by Roebling with his father, John Augustus Roebling, from whom he had gained experience building wire-rope suspension bridges. Upon his father's death, he superintended the building of the Brooklyn Bridge (1869-83). He was disabled by decompression sickness after entering a caisson in 1872. He was brought out nearly insensible and his life was saved with difficulty. Because of resulting poor health, he directed operations from his home in Brooklyn overlooking the site. Though he continued to head the family's wire-rope manufacturing business for several years, medical problems forced retirement (1888).*TIS

*Artists' conception, by Currier and Ives,
of the bridge while construction was underway, 1872

1896 Yurii Dmitrievich Sokolov (May 26, 1896 – February 2, 1971) was a Soviet Ukrainian mathematician.

Sokolov did research on the n-body problem for nearly 50 years. He summarized his work in the 1951 book Singular trajectories of a system of free material points (Russian). He did research on functional equations and on such practical problems as the filtration of groundwater. He also did research on celestial mechanics and hydromechanics.

Sokolov is also known for 'the averaging method with functional corrections' or 'the Sokolov method'. This method is for finding approximate solutions to differential and integral equations.

Sokolov wrote the book The method of averaging of functional corrections (1967), in which he summaries his many important work. He wrote the book at an elementary level. The first part of the book discusses applications of his method to problems which can be modelled by linear integral equations with constant limits. A number of different sufficient conditions for the approximations to converge and presents error estimates were given. The next three parts of the book first examines the problems which can be modelled by nonlinear integral equations with constant limits and then extend the analysis to the situation where the upper limit is variable. In the final part of the book, Sokolov's methods to integral equations of mixed type are examined. He also presented some generalizations of the method in a number of appendices.

For the rescue of Jewish mathematician Semyon Zukhovitskii during the German occupation of Kiev, Yurii Sokolov and his wife Mariya were entered in the list of Righteous Among the Nations.

1899 Otto Neugebauer (May 26, 1899 – February 19, 1990) historian of ancient and medieval mathematics and astronomy. *VFR

He was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age." *Wik

1951 Sally Kristen Ride (May 26, 1951 – July 23, 2012) was an American astronaut and physicist.Born in Los Angeles, she joined NASA in 1978, and in 1983 became the first American woman and the third woman to fly in space, after cosmonauts Valentina Tereshkova in 1963 and Svetlana Savitskaya in 1982. She was the youngest American astronaut to have flown in space, having done so at the age of 32.

Ride was a graduate of Stanford University, where she earned a Bachelor of Science degree in physics and a Bachelor of Arts degree in English literature in 1973, a Master of Science degree in 1975, and a Doctor of Philosophy in 1978 (both in physics) for research on the interaction of X-rays with the interstellar medium. She was selected as a mission specialist astronaut with NASA Astronaut Group 8, the first class of NASA astronauts to include women. After completing her training in 1979, she served as the ground-based capsule communicator (CapCom) for the second and third Space Shuttle flights, and helped develop the Space Shuttle's robotic arm. In June 1983, she flew in space on the Space Shuttle Challenger on the STS-7 mission. The mission deployed two communications satellites and the first Shuttle pallet satellite (SPAS-1). Ride operated the robotic arm to deploy and retrieve SPAS-1. Her second space flight was the STS-41-G mission in 1984, also on board Challenger. She spent a total of more than 343 hours in space. She left NASA in 1987.

Ride worked for two years at Stanford University's Center for International Security and Arms Control, then at the University of California, San Diego, primarily researching nonlinear optics and Thomson scattering. She served on the committees that investigated the loss of Challenger and of Columbia, the only person to participate in both. Having been married to astronaut Steven Hawley during her spaceflight years and in a private, long-term relationship with former Women's Tennis Association player Tam O'Shaughnessy, she is the first astronaut known to have been LGBT. She died of pancreatic cancer in 2012.

DEATHS

735 Bede ( 672/673 – 26 May 735),(often know as the Venerable Bede) Anglo-Saxon theologian, historian and scholar whose writings established the use of BC and AD with dates. He applied a knowledge of astronomy for the purpose of calculating the correct date for Easter. He found that due to an imperfection in Sosigenes' Julian calendar, that the vernal equinox had slipped to a point three days before the traditional date of 21 Mar. However, no action was taken to make the necessary adjustment in the number of leap years per

millenium until nine centuries later. Bede held that the earth was a sphere. He preserved Pytheas' suggestion relating tides to the phases of the moon, and followed Seleucus' idea that a high tide is a local effect and does not occur everywhere at the same point in time. *TIS

Bede was first buried at the monastery of St. Paul at Jarrow in 735. However, in about 1022, his bones were brought to Durham where they were placed with those of St. Cuthbert in the Choir. In 1370, Bede's remains were moved to a splendid shrine in the Galilee Chapel. This shrine was destroyed during the Reformation in 1540 and Bede's bones were then buried in a grave where the shrine had stood.

Eventually, in 1831, the present tomb, made from polished Carboniferous limestone, was erected over Bede's grave. It has the following simple inscription cut into its surface:

HAC SUNT IN FOSSA BEDAE VENERABILIS OSSA

Translated from the Latin, this means 'In this tomb are the bones of the Venerable Bede' *Religion Facts

First page of a manuscript of that includes Bede’s De ratione temporum, with a miniature portrait of Bede in the initial letter “D”, 12th century, University of Glasgow Library (gla.ac.uk) *Linda Hall Org

1838 Ernest Gaston Joseph Solvay (16 April 1838 – 26 May 1922) was a Belgian chemist, industrialist and philanthropist.

Belgian industrial chemist who invented the Solvay Process (1863), a commercially viable ammonia-soda process for producing soda ash (sodium carbonate), widely used in the manufacture of such products as glass and soap. Although a half-century before, A.J. Fresnel had shown (1811) that sodium bicarbonate could be precipitated from a salt solution containing ammonium bicarbonate, many engineering obstacles had to be overcome. Solvay's successful design used an 80 foot tall high-efficiency carbonating tower in which ammoniated brine trickled down from above and carbon dioxide rose from the bottom. Plates and bubble caps helped create a larger surface over which the two could react forming sodium bicarbonate. *TIS

In 1911, he began a series of important conferences in physics, known as the Solvay Conferences, whose participants included Max Planck, Ernest Rutherford, Maria Skłodowska-Curie, Henri Poincaré, and (then only 32 years old) Albert Einstein. A later conference would include Niels Bohr, Werner Heisenberg, Max Born, and Erwin Schrödinger.*Wik

The portrait of participants to the first Solvay Conference in 1911. Ernest Solvay is the third seated from the left. Solvay was not present at the time the photo was taken, so his photo was cut and pasted onto this one for the official release.

1926 Frank Nelson Cole (September 20, 1861 – May 26, 1926) At the time of his death he was a professor of mathematics at Columbia, but was living in a boarding house, under an assumed name, claiming to be a bookkeeper. The AMS Cole prize in algebra is named after him.*VFR

His main research contributions are to number theory, in particular to prime numbers, and to group theory. In number theory he achieved the distinction of being the first to factor 2⁶⁷ - 1 and he did this using quadratic remainders. In fact

2⁶⁷ - 1 = 147573952589676412927 = 761838257287 × 193707721

which a computer will compute in a few seconds today.

For the story of his dramatic presentation of this see Oct 31, 1903

Édouard Lucas had demonstrated in 1876 that M67 must have factors (i.e., is not prime), but he was unable to determine what those factors were.

His contributions to factoring large numbers was published in 1903. His output of research papers was, however, fairly modest and he published only around 25 papers during his career. These publications include his doctoral dissertation in 1886 and a discussion of the icosahedron in 1887. He published The linear functions of a complex variable in the Annals of Mathematics in 1890 then, between the years 1891 to 1893, he found the complete list of simple groups with orders between 200 and 600. Another publication worth mentioning is The triad systems of thirteen letters which he published in the Transactions of the American Mathematical Society in 1913.*Wik According to a notice in the American Mathematical Monthly, which he had edited for twenty-five years, he died of a heart attack brought on by an infected tooth.

Cole is described by D E Smith:-

As a man Cole was admired by all who penetrated a certain reserve that was natural to him, as an executive he was faithful to every duty, as a teacher he was lavish of the time that he would give to those who proved their worth, and as a friend he was loyal to the last. He loved to take long walks in the country studying trees and wild flowers.

1977 Sir Oliver Graham Sutton CBE FRS (4 February 1903 – 26 May 1977) was a Welsh mathematician and meteorologist, notable particularly for theoretical work on atmospheric diffusion, boundary layer turbulence, and for his direction of the UK Meteorological Office.

From 1926 to 1928 he was a lecturer at University College of Wales in Aberystwyth before joining the UK Meteorological Office as an assistant. He was seconded to Shoeburyness to work on the meteorological effects on gunnery practices and then transferred to Porton Down. There he undertook a project on atmospheric turbulence and diffusion which quantified the effect of meteorological conditions on the distribution of gas at ground level, findings which could not be released until after the war. Whilst working at Porton Down he was put in charge of tests related to Operation Vegetarian, which involved the release of anthrax spores over the uninhabited Gruinard Island as part of a biological warfare project.

When the war ended, he was made Chief Superintendent of the Radar Research and Development Establishment, Malvern, a position he held until 1947, when he was appointed Professor of Mathematics at the Royal Military College of Science, Shrivenham, Wiltshire. He was Director-General of the UK Met Office from 1953 to 1965 and Vice-President of the University College of Wales, Aberystwyth from 1967.

He was elected a Fellow of the Royal Society in March 1949. He was awarded CBE in 1950 for his distinguished scientific services to the government.

He was elected president of the Royal Meteorological Society from 1953 to 1955 and awarded their Symons Gold Medal for 1959. He was knighted in 1955.

In 1958 Sutton was invited to co-deliver the Royal Institution Christmas Lecture. In 1968 he was awarded the prestigious International Meteorological Organization Prize from the World Meteorological Organization *Wik

1984 Mary Taylor Slow (15 July 1898 – 26 May 1984) was a British physicist who worked on the theory of radio waves and the application of differential equations to physics. She was the first woman to take up the study of radio as a profession.

Mary Taylor was born in Sheffield, England. Both her parents were schoolteachers. She was educated at Pomona Street Elementary School in Sheffield and then Sheffield High School, from which she won a Clothworker's Scholarship to Girton College, Cambridge. She studied the Natural Sciences Tripos; in 1919 she was awarded the equivalent of a first-class BA degree, and in 1920 she graduated in mathematics and natural sciences.

Taylor continued to study at Girton College under a series of research studentships. From 1922 to 1924 she was Assistant Lecturer in Mathematics at Girton. During this time she became interested in the theory of radio waves and started to conduct research under the guidance of Edward Appleton who was then assistant demonstrator in experimental physics at the Cavendish Laboratory in Cambridge.

When Appleton left Cambridge to join King's College, London, Taylor moved from Cambridge to the University of Göttingen in Germany. Here she was awarded her PhD in 1926 for a thesis on aspects of electromagnetic waves that she wrote in German. Taylor was awarded a Yarrow Research Fellowship which enabled her to remain at Göttingen and continue her work on electromagnetic waves with Professor Richard Courant.

In 1929 Taylor returned to the UK and took up a post as Scientific Officer at the Radio Research Station in Slough, Berkshire (part of the UK Department of Scientific and Industrial Research and the UK National Physics Laboratory, now the National Physical Laboratory). Here she continued to carry out research into the theory of electromagnetic waves, specializing in the magneto-ionic theory of radio wave propagation and in the application of differential equations to physics and radio. During this period she published two papers in the Proceedings of the Physical Society, both on aspects of the Appleton-Hartree Equation. Taylor was a member of the London Mathematical Society and the Cambridge Philosophical Society.

2003 Gerald Stanley Hawkins ( April 28, 1928 Gt. Yarmouth, Norfolkshire, U.K. - May 26, 2003) was a British-born American radio-astronomer who used a computer to show that the stones and other archaeological features at Stonehenge formed a pattern of alignments with 12 major lunar and solar events, suggesting that it was used as a sort of neolithic observatory or astronomical calendar. In the 18th century, William Stukely had noticed that the horseshoe of trilithons and 19 bluestones opened up in the direction of the midsummer sunrise. In the 1960s, Hawkins, a British-born radio astronomer, identified 165 key points in the neolithic complex and found that many were strongly correlated with the rising and setting positions of the sun and moon over an 18.6-year cycle. In the 1990s, he studied the geometry of crop circles. He retired to a Virginia farm in Rappahannock County with his second wife, Julia Dobson, and died there, suddenly, on May 26, 2003.