Pat'sBlog

Friday, 3 July 2026

On This Day in Math - July 3

Music is the pleasure the human mind experiences from counting

without being aware that it is counting.

~Gottfried Leibniz

The 184th day of the year; 184 = 23 * 2³ (concatenation of the first two primes).

The smallest number that can be written as q * p^q + r * p ^r, where p, q and r are distinct primes (184 = 3 * 2³ + 5 * 2⁵). *Prime Curios

25^2-21^2 = 184, and the sum of three squares 12^2 + 6^2 + 2^2 and of four squares, 10^2 + 8^2+4^2 + 2^2

On a 5x5 lattice (square grid of dots) there are 184 paths from one corner to the opposite corner touching each lattice point exactly once.

The concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year days)

EVENTS

1822 Charles Babbage described his ideas for a “difference engine” to the Royal Society. *VFR

1841, John Couch Adams decided to determine the position of an unknown planet by the irregularities it causes in the motion of Uranus. He entered in his journal; "Formed a design in the beginning of this week in investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus... in order to find out whether they may be attributed to the action of an undiscovered planet beyond it..." In Sep 1845 he gave James Challis, director of the Cambridge Observatory, accurate information on where the new planet, as yet unobserved, could be found; but unfortunately the planet (Neptune) was not recognized at Cambridge until much later, after its discovery at the Berlin Observatory on 23 Sep 1846. *TIS

Using predictions made by Urbain Le Verrier, Johann Galle discovered the planet in 1846. The planet is named after the Roman god of the sea, as suggested by Le Verrier.

It turned out that several astronomers, starting with Galileo Galilei in 1612, had observed Neptune too, but because of its slow motion relative to the background stars. did not recognize it as a planet.

*NASA

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1886 Karl Benz officially unveils the Benz Patent-Motorwagen, the first purpose-built automobile. *

the painter flynn

2011 Astronomers using the Hubble Space Telescope discovered a fourth moon orbiting the icy dwarf planet Pluto. The tiny, new satellite – temporarily designated P4 -- was uncovered in a Hubble survey searching for rings around the dwarf planet.

Two labeled images of the Pluto system taken by the Hubble Space Telescope's Wide Field Camera 3 ultraviolet visible instrument with newly discovered fourth moon P4 circled. The image on the left was taken on June 28, 2011. The image of the right was taken on July 3, 2011. Credit: NASA, ESA, and M. Showalter (SETI institute)

BIRTHS

1820 Ernest de Jonquières (3 July 1820 Carpentras, France – 12 Aug 1901 Mousans-Sartoux, France) was a French naval officer who discovered many results in geometry. After his introduction to advanced mathematics by Chasles it is not surprising that his main interests were geometry throughout his life. He made many contributions many of them extending the work of Poncelet and Chasles. An early work, the treatise Mélanges de géométrie pure (1856) contains: an amplifications of Chasles' ideas on the geometric properties of an infinitely small movement of a free body in space; a commentary on Chasles' work on conic sections; the principle of homographic correspondence; and constructions relating to curves of the third order. In a final section de Jonquières presented a French translation of Maclaurin's work on curves. *SAU

1849 Prosper-René Blondlot (3 July 1849 – 24 November 1930) was a French physicist, best remembered for his mistaken "discovery" of N rays, a phenomenon that subsequently proved to be illusory.
In order to demonstrate that a Kerr cell responds to an applied electric field in a few tens of microseconds, Blondlot, in collaboration with Ernest Bichat, adapted the rotating-mirror method that Léon Foucault had applied to measure the speed of light. He further developed the rotating mirror to measure the speed of electricity in a conductor, photographing the sparks emitted from two conductors, one 1.8 km longer than the other and measuring the relative displacement of their images. He thus established that the speed of electricity in a conductor is very close to that of light.
In 1891, he made the first measurement of the speed of radio waves, by measuring the wavelength using Lecher lines. He used 13 different frequencies between 10 and 30 MHz and obtained an average value of 297,600 km/s, which is within 1% of the current value for the speed of light. This was an important confirmation of James Clerk Maxwell's theory that light was an electromagnetic wave like radio waves.
In 1903, Blondlot announced that he had discovered N rays, a new species of radiation. The "discovery" attracted much attention over the following year until Robert W. Wood showed that the phenomena were purely subjective with no physical origin. ( Wood had a reputation for debunking in that period.) The French Academy of Sciences awarded the Prix Leconte (₣50,000) for 1904 to Blondot, although they hedged on the reason, citing the totality of his work rather than the discovery of N-rays.
Little is known about Blondlot's later years. William Seabrook stated in his Wood biography Doctor Wood, that Blondlot went insane and died, supposedly as a result of the exposure of the N ray debacle: "This tragic exposure eventually led to Blondlot's madness and death." Using an almost identical wording this statement was repeated later by Martin Gardner, possibly without having investigated into the subject: "Wood's exposure led to Blondlot's madness and death." However, Blondlot continued to work as a university professor in Nancy until his early retirement in 1910. He died at the age of 81; at the time of the N-ray affair he was nearly 60 years old. *Wik

1866 Henry Frederick Baker FRS (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as solitons), and Lie groups.
Baker was born in Cambridge, England and educated at The Perse School before winning a scholarship to St John's College, Cambridge in October 1884. Baker graduated as Senior Wrangler in 1887, bracketed with 3 others.
Baker was elected Fellow of St John's in 1888 where he remained for 68 years.
In June, 1898 he was elected a Fellow of the Royal Society. In 1911, he gave the presidential address to the London Mathematical Society.
In January 1914 he was appointed Lowndean Professor of Astronomy. *Wik
In the 1930s before the war Baker's graduate students would meet at what they called Professor Baker’s "Tea Party", who met once a week to discuss the areas of research in which we were all interested. It was to one of these meetings that a young Donald Coxeter brought his "Aunt Alice", the 69 year old Alicia Boole, to co-present on the subject of Polytopes in higher dimensions.

1908 Archibald James Macintyre HFRSE (3 July 1908 – 4 August 1967) was a British-born mathematician.

He was born in Sheffield on 3 July 1908, the second child of William Ewart Archibald Macintyre (b.1878) previously of Long Eaton, and his wife, Mary Beatrice Askew. His father was a schoolmaster in Sheffield and his mother was a former teacher.

Archibald was educated at the Central Secondary School in Sheffield (previously known as the High Storrs Grammar School). He left school in 1926 and won a place at Magdalene College, Cambridge studying a Mathematics Tripos under Arthur Stanley Ramsey. Fellow students included Donald Coxeter, Raymond Paley and Harold Davenport. He graduated BA as a Wrangler in 1929 then began research under Dr Edward Collingwood.

In 1930 he became an assistant lecturer in both applied maths and theoretical physics at Cambridge University. He received his doctorate (PhD) in 1933. In 1936 he accepted a post of lecturer at Aberdeen University. Here he stayed for many years, rising to senior lecturer. In 1947 he was elected an Honorary Fellow of the Royal Society of Edinburgh. His proposers were E. M. Wright, Ivor Etherington, Edward Thomas Copson, Edmund Taylor Whittaker and James Cossar.

In 1958 he moved to the University of Cincinnati in the United States, as a visiting professor of mathematics. He was recruited primarily as a reaction to Sputnik. America wanted to increase its role in the sciences and math. His wife stayed in Aberdeen, Scotland where she continued to teach mathematics at King's College. A year later he accepted a permanent position at the University of Cincinnati and sent for his wife who was also given a teaching position as a lecturer in mathematics. They formed a highly unusual husband-wife team.

He died in Cincinnati on 4 August 1967, eight years after his wife died of breast cancer

1910 Antoine Joseph Bernard Brunhes (3 July 1867 – 10 May 1910) was a French geophysicist known for his pioneering work in paleomagnetism, in particular, his 1906 discovery of geomagnetic reversal. The current period of normal polarity, Brunhes Chron, and the Brunhes–Matuyama reversal are named for him.

1897 Jesse Douglas (3 July 1897 – 7 September 1965) born in New York City. He did important work on Plateau’s problem, which asks for the minimal surface connecting a given boundary. For this work he received a Fields medal in 1936, the ﬁrst time that they were given. *VFR ..the Plateau problem... had first been posed by the Italian-French mathematician Joseph-Louis Lagrange in 1760. The Plateau problem is one of finding the surface with minimal area determined by a fixed boundary. Experiments (1849) by the Belgian physicist Joseph Plateau demonstrated that the minimal surface can be obtained by immersing a wire frame, representing the boundaries, into soapy water. Douglas developed what is now called the Douglas functional, so that by minimizing this functional he could prove the existence of the solution to the Plateau problem. Douglas later developed an interest in group theory.*TIS

1913 Bibha Chowdhuri (3 July 1913 – 2 June 1991) was an Indian particle physicist known for her investigations into cosmic rays. Working with D. M. Bose, she was the first to discover mesons and proving Hideki Yukawa mesons theory. [ Chowdhuri didn't "discover" mesons alone, but she, with D.M. Bose, published pioneering research around 1942-1944, providing crucial evidence for the pi-meson (π-meson) using cosmic rays and photographic plates, work that paved the way for C.F. Powell's Nobel-winning discovery later. Her team's experiments showed particles with decaying mass, suggesting new particles, though they lacked resources for full confirmation, and Powell later credited their early findings. ]

Chowdhuri demonstrated that the density of penetrating events is proportional to the total particle density of an extensive air shower. She was interviewed by The Manchester Herald in an article called "Meet India's New Woman Scientist – She has an eye for cosmic rays", saying that "it is a tragedy that we have so few women physicists today."

Chowdhuri's cosmic ray studies contributed heavily to the discovery of K mesons. Bibha temporarily left TIFR in 1953 and subsequently joined cosmic ray physicist L. Leprince Ringuet’s lab under the Centre National de la Recherche Scientifique (Paris). She studied and identified many new K mesons in cloud chambers on the Alps, publishing the research in the Nuovo Cimento in 1957. In 1954 she was a visiting researcher at the University of Michigan. She was appointed because Homi Bhabha was still establishing the Tata Institute of Fundamental Research, and contacted her thesis examiners for advice on outstanding graduate students. She joined the Physical Research Laboratory and became involved with the Kolar Gold Fields experiments. She moved to Kolkata to work at the Saha Institute of Nuclear Physics. She taught physics in French She continued to publish until she died in 1991.*Wik

1933 Frederick Justin Almgren,(July 3, 1933, Birmingham, Alabama–February 5, 1997, Princeton, New Jersey) Almost certainly Almgren's most impressive and important result was only published in 2000, three years after his death. Why was this? The paper was just too long to be accepted by any journal. Brian Cabell White explains the background in a review of the book published in 2000 containing the result:-
By the early 1970s, geometric analysts had made spectacular discoveries about the regularity of mass-minimizing hypersurfaces. (Mass is area counting multiplicity, so that if k sheets of a surface overlap, the overlap region is counted k times.) In particular, the singular set of an m-dimensional mass-minimizing hypersurface was known to have dimension at most m - 7. By contrast, for an m-dimensional mass-minimizing surface of codimension greater than one, the singular set was not even known to have m-measure 0. Around 1974, Almgren started on what would become his most massive project, culminating ten years later in a three-volume, 1700-page preprint containing a proof that the singular set not only has m-dimensional measure 0, but in fact has dimension at most (m - 2). This dimension is optimal, since by an earlier result of H Federer there are examples for which the dimension of the singular set is exactly (m - 2). ... Now, thanks to the efforts of editors Jean Taylor and Vladimir Scheffer, Almgren's three-volume, 1700-page typed preprint has been published as a single, attractively typeset volume of less than 1000 pages.*SAU

1933 William (Bill) Parry FRS (3 July 1934–20 August 2006) was an English mathematician. During his research career, he was highly active in the study of dynamical systems, and, in particular, ergodic theory, and made significant contributions to these fields. He is considered to have been at the forefront of the introduction of ergodic theory to the United Kingdom. He played a founding role in the study of subshifts of finite type, and his work on nilflows was highly regarded.*Wik

1945 Saharon Shelah (July 3, 1945 - ) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Shelah is one of the most prolific contemporary mathematicians. As of 2009, he has published nearly 900 mathematical papers (together with over 200 co-authors).

His main interests lie in mathematical logic, model theory in particular, and in axiomatic set theory. In model theory, he developed classification theory, which led him to a solution of Morley's problem. In set theory, he discovered the notion of proper forcing, an important tool in iterated forcing arguments. With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N). He extended Arrow's impossibility theorem on voting systems. *Wik

DEATHS

1749 William Jones, FRS (1675 – 3 July 1749) was a Welsh mathematician, most noted for his proposal for the use of the symbol π (the Greek letter pi) to represent the ratio of the circumference of a circle to its diameter. He was a close friend of Sir Isaac Newton and Sir Edmund Halley. In November, 1711 he became a Fellow of the Royal Society, and was later its Vice-President.*Wik

Jones Pi for C/d *Wik

1789 Jakob Bernoulli II died. *VFR Born in Basel in 1759 (17 October), the son of Johann (II). assistant to Daniel in experimental physics

He graduated in Jurisprudence in 1778 but also studied Maths and Physics. In 1782, he applied for Daniel's former chair but was unsuccessful.

He became secretary to an imperial representative in Venice

In 1786 he went to Petersburg, to the Academy of Science (Fuss recommended him to Dashkoff) and in 1788 became ordentlich academy member for mathematics.

He married one of Euler's granddaughters, Charlotte.

At thirty years of age, he drowned in the Neva. *Brian Daugherty

In St Petersburg he began to write important works on mathematical physics which he presented to the St Petersburg Academy of Sciences. These treatises were on elasticity, hydrostatics and ballistics.

Despite the rather harsh climate, the city of St Petersburg had great attractions for Jacob(II) Bernoulli since his uncle Daniel Bernoulli had worked there with Euler. In fact Jacob(II) married a granddaughter of Euler in St Petersburg but, tragically, the city was to lead to his death.

St Petersburg is located on the delta of the Neva River, at the head of the Gulf of Finland. St Petersburg, built on 42 islands in the Neva River, is a city of waterways and bridges and because of this it is called the "Venice of the North." Bernoulli drowned, while still only 29 years of age, in the Neva River while he was swimming.

1970 Samuel Beatty (Aug 21, 1881– 3 Jul, 1970) was a Canadian mathematician who was the first person to receive a PhD in mathematics from a Canadian university.

He entered the University of Toronto in 1903 as an undergraduate. He was to spend the rest of his life studying there and working for the University. After obtaining his undergraduate degree from Toronto, Beatty went on the undertake research for a Ph.D. under Fields's supervision. When Beatty was awarded a doctorate in 1915 he became the first to obtain such a degree from a Canadian university. In fact Beatty was the only student who Fields supervised for a doctorate.

Beatty was appointed as a Lecturer at the University of Toronto after studying for his doctorate. When he was appointed, Alfred Baker was his Head of the Mathematics Department, but in 1918 Baker retired and A T DeLury, who had taught Beatty when he was an undergraduate, became Head. Beatty was promoted to Professor, then in 1934 became Head of the Mathematics Department. In 1936, in addition to his role has Head of the Mathematics Department, he was appointed Dean of the Faculty of Arts and, three years later became a founding member of the Committee of Teaching Staff.

He retired from the role of Dean in 1952 and in the following year was elected Chancellor of the University. He held this position until 1959. First let us quote an episode relating to his time as Dean:-

Dean Beatty is remembered for the enormous support he gave to his students, and he earned their deepest appreciation as a result. One of his students, Walter Kohn, who won the 1998 Nobel Prize in Chemistry for his development of the density-functional theory, expressed heartfelt appreciation to the Dean who in 1942 helped Kohn to enrol in the Mathematics Department at the University. Kohn, a young chemist of enormous potential, could not gain access to the chemistry buildings during the war because of his German nationality, and Dean Beatty was instrumental in helping him to continue his studies.

1991 Ernst Witt (June 26, 1911-July 3, 1991) was a German mathematician born on the island of Als (German: Alsen). Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine.
Witt's work was mainly concerned with the theory of quadratic forms and related subjects such as algebraic function fields.

Witt's work has been highly influential. His invention of the Witt vectors clarifies and generalizes the structure of the p-adic numbers. It has become fundamental to p-adic Hodge theory.

Witt was the founder of the theory of quadratic forms over an arbitrary field. He proved several of the key results, including the Witt cancellation theorem. He defined the Witt ring of all quadratic forms over a field, now a central object in the theory.

The Poincaré–Birkhoff–Witt theorem is basic to the study of Lie algebras. In algebraic geometry, the Hasse–Witt matrix of an algebraic curve over a finite field determines the cyclic étale coverings of degree p of a curve in characteristic p.

In the 1970s, Witt claimed that in 1940 he had discovered what would eventually be named the "Leech lattice" many years before John Leech discovered it in 1965, but Witt did not publish his discovery and the details of exactly what he did are unclear.*Wik

1999 Pelageya Yakovlevna Polubarinova-Kochina (13 May 1899[O.S.] – 3 July 1999) was a Soviet and Russian applied mathematician, known for her work on fluid mechanics and hydrodynamics, particularly, the application of Fuchsian equations [ A linear differential equation in which every singular point, including the point at infinity, is a regular singularity], as well as in the history of mathematics. She was elected a corresponding member of the Academy of Sciences of the Soviet Union in 1946 and full member (academician) in 1958.

She studied at a women's high school in Saint Petersburg and went on to Petrograd University (after the Russian Revolution). After her father died in 1918, she started working at the laboratory of geophysics under the supervision of Alexander Friedmann. There she met Nikolai Kochin; they were married in 1925 and had two daughters. The two taught at Petrograd University until 1934, when they moved to Moscow, where Nikolai Kochin took a teaching position at the Moscow University. In Moscow, Polubarinova-Kochina did research at the Steklov Institute until World War II, when she and their daughters were evacuated to Kazan while Kochin stayed in Moscow to work on aiding the military effort. He died before the war was over.

After the war, she edited his lectures and continued to teach applied mathematics. She was later head of the department of theoretical mechanics at the University of Novosibirsk and director of the department of applied hydrodynamics at the Hydrodynamics Institute. She was one of the founders of the Siberian Branch of the Russian Academy of Sciences (then the Academy of Sciences of the Soviet Union) at Novosibirsk.

She was awarded the Stalin Prize in 1946, was made a Hero of Socialist Labour in 1969 and received the Order of Friendship of Peoples in 1979. She died in 1999, a few months after her 100th birthday, and shortly after publishing her last scientific article.*Wik

2017 Albert "Tommy" Wilansky (13 September 1921, St. John's, Newfoundland – 3 July 2017, Bethlehem, Pennsylvania) was a Canadian-American mathematician, known for introducing Smith numbers.

Wilansky was educated as an undergraduate at Dalhousie University, where he received an M.A. in mathematics in 1944. From 1944 to 1947 he was a graduate student at Brown University. In 1947 he received his Ph.D. with advisor Clarence Raymond Adams and dissertation An application of Banach linear functionals to the theory of summability.

From 1948 until his official retirement in 1992, Wilansky was a faculty member of the mathematics department of Lehigh University.

He was the university’s Distinguished Professor of Mathematics for the final 14 years of his tenure. During his 44 years at Lehigh he was a Fulbright visiting professor several times, at universities in Reading (1972–1973), London (1973), Tel Aviv (1981), and Berne (1981). Outside of academia he was a consultant for the Frankford Arsenal for the year 1957–1958.

Wilansky did research in analysis, specializing in summability theory, linear topological spaces, Banach algebras, and functional analysis. He was the author of several books and the author or co-author of more than 80 articles. He lectured at over 50 different universities. In 1969 he received the Mathematical Association of America's Lester R. Ford Award for his 1968 article Spectral Decomposition of Matrices for High School Students. *Wik

In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.

Smith numbers were so-named by Wilansky as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:

4937775 = 3 · 5 · 5 · 65837 while 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7) in base 10.

Other Smith numbers are 4, 22, 27, 58, 85, 121, 166..... (OEIS A006753)

2022 Robert Floyd Curl Jr. (August 23, 1933 – July 3, 2022) American chemist who with Richard E. Smalley and Sir Harold W. Kroto discovered the first fullerene, a spherical cluster of carbon atoms, in 1985. The discovery opened a new branch of chemistry, and all three men were awarded the 1996 Nobel Prize for Chemistry for their work. In Sep 1985 Curl met with Kroto of the University of Sussex, Eng., and Smalley, a colleague at Rice, and, in 11 days of research, they discovered fullerenes. They announced their findings to the public in the 14 Nov 1985, issue of the journal Nature.*TIS

Born in Alice, Texas, United States, Curl was the son of a Methodist minister. Due to his father's missionary work, his family moved several times within southern and southwestern Texas, and the elder Curl was involved in starting the San Antonio Medical Center's Methodist Hospital.[Curl attributes his interest in chemistry to a chemistry set he received as a nine-year-old, recalling that he ruined the finish on his mother's porcelain stove when nitric acid boiled over onto it.

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

Oldest Multiplication Table Found?

"From an online article in Nature, Jan 2014:
From a few fragments out of a collection of 23-century-old bamboo strips, historians have pieced together what they say is the world's oldest example of a multiplication table in base 10.
(The Egyptian scrolls of 17th Century BC give a method equivalent to multiplication in base two ,although the scrolls are generally in base 60, see below)

Five years ago, Tsinghua University in Beijing received a donation of nearly 2,500 bamboo strips. Muddy, smelly and teeming with mould, the strips probably originated from the illegal excavation of a tomb, and the donor had purchased them at a Hong Kong market. Researchers at Tsinghua carbon-dated the materials to around 305 bc, during the Warring States period before the unification of China.

Each strip was about 7 to 12 millimetres wide and up to half a metre long, and had a vertical line of ancient Chinese calligraphy painted on it in black ink. Historians realized that the bamboo pieces constituted 65 ancient texts and recognized them to be among the most important artefacts from the period.

When the strips are arranged properly, says Feng, a matrix structure emerges. The top row and the rightmost column contain, arranged from right to left and from top to bottom respectively, the same 19 numbers: 0.5; the integers from 1 to 9; and multiples of 10 from 10 to 90.

The researchers suspect that officials used the multiplication table to calculate surface area of land, yields of crops and the amounts of taxes owed. “We can even use the matrix to do divisions and square roots,” says Feng. “But we can’t be sure that such complicated tasks were performed at the time.”

The table didn't show particular

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I checked with AI to see if there had been any up date on the story and got a little more information. I also wonder how I didn't pick up on the 1/2 in the multiplication table. One of the things tht make this table important was that before this, it seems they never offered up a calculation assistant like this. Prior to this they might give a single example of two, like 7 x 8= 56 or 3 x 9 = 27. This table was different in that it allowed the clerk to take any quantity and break nit down by the same multiplication and addition method students learn today. The tables sort of looked like :(my creation)

---- 1/2 1 2 3 4 5 ...

1 1/2 1 2 3..

2 4 6...

3 9 ...

10 - 5 10 20 30

20 - 10 20 40

30 - 15 30

40 -

From the strips that are preserved it seems like they only showed one triangle of the tables, since 6 x 9 is the same as 9 x 6, they didn't show any products in forms like 24 x 32. They would break out the numbers into 20 x 30 + 20 x 2 + 4 x 30 + 4 x 2.

An older multiplication method (although not technically a multiplication table) is presented in ancient Egyptian scrolls)

In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, was a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (generally the larger) into a sum of powers of two and creates a table of doublings of the second multiplicand. This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas.

The name for most of history, and still a good name today, is still mediation and duplation. The names like Russian Peasant multiplication seem not to have occurred much before 1910. My take is someone wanted to show it to students and began with something like "a method used in poor countries where they didn't get formal education,," and then slid into the Russian Peasant Multiplication

The method worked by makin a table starting with the values to be multiplied, let's use 42 x 47

when we halve the number if it is odd, we reduceit by one and then take half

Double Half Odd or Not

42 47 odd

84 23 odd

168 11 odd

336 5 odd

672 2 Not Odd

1344 1 Odd

Now ignore all the left column values if the half value is even, and add all the rest... 42 + 84 + 168 + 336 + 1344. The total is 1974.

Notice that the numbers in the left table are all multiples of 42.

We have 42 x 1, 42 x 2, 42 x 4, 42 x 8, 42 x 16, and 42 x 32. We left out the row for 42 x 16, and added the rest. So how many 42's did we add, 1+2+4+8+ 32.... add em up, 47. Working backwards we get the binary number for 47, 101111.

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The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.

Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors. *wik

This tablet, from about 2000 BC, was a school math book for teaching kids how to calculate inheritance. The problem asks how much each of seven boys would get when their father died, according to Babylonian law. Apparently the law said they should each get a different proportion, with the oldest getting the most and the younger kids less and less. Whoever did the math worked up from the bottom (which was not normal), and also made a mistake in his or her calculations!

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Maybe I'll think of more to go here later....or one of you bright people will send me an extra bit I didn't know.

Thursday, 2 July 2026

On This Day in Math - July 2

OOPS...

Heavier-than-air flying machines are impossible.
I have not the smallest molecule of faith in aerial navigation other than ballooning,
or of the expectation of good results from any of the trials we hear of. 1895
And in a Letter to Baden-Powell (1896) Radio has no future.

William Thomson, Lord Kelvin

The 183rd day of the year; the concatenation of 183 and 184, 183184 is a perfect square. There are no smaller numbers for which the concatenation of two consecutive numbers is square. (Students might seek the next such pair of numbers. They are small enough to be year dates)

The sum of the first 183 primes minus 183 is prime.

183 is the difference of two squares, 32^2 - 29^2, and of course, like every odd number, it is the difference of the squares ot the consecutive numbers that sum to 183, 92^2-91^2= 183

Lagrange proved that every integer is the sum of four or less non-zero squares. 183 is one of the unusual ones that require four. It is the 29th number that requires the full set of three squares. $183 = 13^2 + 3^2 + 2^2 + 1^1.$

183 is the eighth of the 12 year-days which are perfect totient numbers. (There are only 57 such numbers under 10³). A list of the perfect totient numbers seems to suggest that all of them are multiples of three, but then you get to 4375, the smallest perfect totient number that is not divisible by 3.[a perfect totient number is a number that is the sum of it's iterated totients, that is, the number of integers smaller than, and relatively prime to 183 + the number smaller than and less than that result, + ... down to one, "For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327 " *Wik
Get more Math Facts for every year date here

EVENTS

1133 First trade security agreement between Pisans and Alibibn Yusof of Morocco.Under such an agreement, Guilielmo, father of Leonardo brought his young son (who would much later be called Fibonacci) to Bugia, where he would learn of the Arabic calculation system that he would introduce to his homeland in his Liber Abbaci in 1202. *Devlin, The Man of Numbers

In 1698 Thomas Savery was issued British Letters Patent No.356 for what he called "The Miner's Friend; or an engine to raise water by fire" which was the first application of steam power for pumping water. The steam powered pump had no piston. Two years earlier Savery was issued British Letters Patent No.347 for his invention for "Navigation improved; or the art of rowing ships of all rates in calms with a more easy, swift, and steady motion than oars can." ( which involved paddle-wheels driven by a capstan and which was dismissed by the Admiralty following a negative report by the Surveyor of the Navy,Edmund Dummer.)*TIS*Wik

1779 One of the earliest mentions of blackboards in Colonial America was in a letter from John Taylor, a tutor at Queens College (to become Rutgers University)to a graduate student, John Bogart, that he was asking to take over his classes while Taylor was away on military duties. "I have spoken to Mr. Briton to make a blackboard.." *Kidwell, Hastings, & Roberts; Tools of American Mathematics Teaching.

1832 Legendre writes to Nathanial Bowditch regarding his translation of LaPlace's "Mecanique Celeste" ,"Your work is not merely a translation with a commentary; I regard it as a new edition, augmented and improved, and such a one as might have come from the hands of the author himself, ... if he had been solicitously studious of being clear." LaPlace's classic is a very difficult book, and Biot, who helped him prepare it for printing said that Laplace himself would frequently get lost in following his own line of reasoning and insert, "il est aise a voir" (It is easy to see). *The Teaching and History of Mathematics in the United States, F. Cajori.

At this time Bowditch was regarded as perhaps the only world class mathematicians of the new continent.

1850 Stokes’s theorem made its ﬁrst appearance as a postscript to a letter from Sir William Thompson (Lord Kelvin) to Stokes. It ﬁrst appeared publicly as question 8 on the Smith’s Prize exam for 1854. Stokes drew up this competitive exam, which was taken by the best mathematics students at Cambridge University. By the time Stokes died the theorem was universally known as “Stokes’s theorem.” [Spivak, Calculus on Manifolds, p. viii].

1865 Sylvia Ann Howland died in 1865, leaving roughly half her fortune of some 2 million dollars (equivalent to $31,291,000 in 2016) to various legatees, with the residue to be held in trust for the benefit of Hetty Robinson, Howland's niece. The remaining principal was to be distributed to various beneficiaries on Robinson's death.

Robinson produced another will, leaving her the whole estate outright. To the will was attached a second and separate page, putatively seeking to invalidate any subsequent wills. Howland's executor, Thomas Mandell, rejected Robinson's claim, insisting that the second page was a forgery, and Robinson sued.

In the ensuing case of Robinson v. Mandell, Charles Sanders Peirce testified that he had made pairwise comparisons of 42 examples of Howland's signature, overlaying them and counting the number of downstrokes that overlapped. Each signature featured 30 downstrokes and he concluded that, on average, 6 of the 30 overlapped, 1 in 5. Benjamin Peirce showed that the number of overlapping downstrokes between two signatures also closely followed the binomial distribution, the expected distribution if each downstroke was an independent event. When the admittedly genuine signature on the first page of the contested will was compared with that on the second, all 30 downstrokes coincided, suggesting that the second signature was a tracing of the first.

Benjamin Peirce, Charles' father, then took the stand and asserted that, given the independence of each downstroke, the probability that all 30 downstrokes should coincide in two genuine signatures was

\(\frac {1}{2.666* 10^{21}}/) *wik

C S Peirce

1866 Alfred Russell Wallace writes to Darwin with to suggest a name change for his basic principal of Evolution:

I wish.. to suggest ... adopting Spencer's term, (which he generally uses in preference to Natural Selection, viz, "Survival of the Fittest"

*Mario Livio, Brilliant Blunders, pg 29

1883 Helmholtz writes to Heinrich Hertz to congratulate him on his investigations. "I have read with the greatest interest your investigation on the cathode ray discharge, and cannot refrain from writing to say Bravo!" Hemholtz was not given to token praise, and was the opinion that was most valued by Hertz. *Hertz Miscellaneous Papers

Cathode Ray Tube or Crookes Tube

1897 – Italian scientist Guglielmo Marconi obtains a patent for radio in London.*Wik

1944 Grace Hopper meets Howard Aiken for the first time. Here is her description of the meeting:

Until 1944, I had been a thoroughly respectable mathematician. I had never met a digit, and I wanted nothing to do with digits. I came into the computer business in a unique fashion. I was ordered to the Navy Liaison Officer at Harvard. I left Midshipmen School on Friday, and on Monday morning, 2nd July 1944, I reported to the Navy Liaison Officer, Harvard. He took me by the hand, and led me over to an underground laboratory. I had just acquired one-and-a-half stripes. There stood a large object, with three stripes, who took one look at me and said: "Where the hell have you been?" I spluttered, and said that I had had two days' travel time. He said: "For the last three months." I said: "Midshipmen's School", and he said he had told them it was not necessary. By this time I was practically cowering, of course, but with one-and-a-half stripes, I would stand up straight and listen to three stripes. He waved his hand and said: "That's a computing machine." I said, "Yes, Sir." What else could I say? He said he would like to have me compute the coefficients of the arc tangent series, for Thursday. Again, what could I say? "Yes, Sir." I did not know what on earth was happening, but that was my meeting with Howard Hathaway Aiken. In the long run, he taught me one very important thing. One can always make a mistake once, but it must not be made a second time. That was a very good thing to learn. He also flatly informed me that he had told the Bureau of Naval Personnel not to send him a female. At this point, and over the next few months, I learned another lesson. I could have taken the attitude of showing him, and making him take that back. Instead, I decided it would be far better to learn to work with him. This is a lesson we all need to learn: of not showing people, but learning to work with people. It certainly made a difference in getting things done in the computer field.

*SAU

1971 On this day in 1971, the first meeting of the British Society for the History of Mathematics took place at Thames Polytechnic [now the University of Greenwich].

1982 Science (p. 39) reported that Steven Smale had proved that the average-case behavior of the simplex algorithm for linear programming is far better than the worse-case behavior, which is exponential. [Mathematics Magazine 56 (1983), p. 55]. *VFR

1992 On July 2, fearing for the impact that a park service removal of homeowners around Elkmont, Tennessee in the Great Smokey Mountain Natl Park would have on a variety of local fireflys, Lynn Faust wrote a letter to Professor Steven Stogratz, who had recently written a paper in Science Magazine about synchronized flashing of the firelies (lightning bugs) known to exist in Southeast Asia. Ms. Faust's letter would, in Strogatz's words, "shatter a (scientific) myth that had lasted for decades."

2011 Simon Chua, 58, received Australia's B. H. Neumann Award for pioneering efforts in training and honing the talent of young Filipino mathematicians for the past 15 years.
The Australian Mathematics Trust (AMT) bestowed the award to Chua on July 2, 2011. The B. H. Neumann Awards are presented annually for "important contributions over many years to the enrichment of mathematics learning in Australia and its region," according to the AMT.
The award is named in honor of Bernhard H. Neumann, the so-called father of Australian mathematics, who “provided outstanding leadership, support and encouragement for mathematics and the teaching of mathematics at all levels."
Chua, who is president and co-founder of the Mathematics Trainers Guild-Philippines (MTG), is the first Asian to receive the award.
Through the MTG, Filipino students have won numerous medals in math competitions abroad including in China, United States, Singapore, Thailand, Hong Kong and Indonesia.
In 2006, Chua became the first Filipino to win the Paul Erdos Award from the World Federation of National Mathematics Competitions. *MathDL

Simon Chua receives his Erdös Award from WFNMC President Peter Kenderov at Robinson College, Cambridge on 22 July 2006.

BIRTHS

1622 Rene-Francoise de Sluse (2 July 1622 – 19 March 1685) Sluse contributed to the development of calculus and this work focuses upon spirals, tangents, turning points and points of inflection. He and Johannes Hudde found algebraic algorithms for finding tangents, minima and maxima that were later utilized by Isaac Newton. These algorithms greatly improved upon the complicated algebraic methods of Pierre de Fermat and René Descartes, who themselves had improved upon Roberval's kinematic, but geometric, non-algorithmic methods of determining tangents. Augustus De Morgan has the following to say about de Sluse's contribution to Newton's method of fluxions in his discussion of the Leibniz–Newton calculus controversy.

When they state that Collins had been four years in circulating the letter in which the method of fluxions was sufficiently described to any intelligent person, they suppress two facts: first, that the letter itself was in consequence of Newton's learning that Sluse had a method of tangents; secondly, that it revealed no more than Sluse had done. ...this method of Sluse is never allowed to appear ...Sluse wrote an account of the method which he had previously signified to Collins, for the Royal Society, for whom it was printed. The rule is precisely that of Newton... To have given this would have shown the world that the grand communication which was asserted to have been sent to Leibniz in June 1676 might have been seen in print, and learned from Sluse, at any time in the previous years: accordingly it was buried under reference. ...Leibniz had seen Hudde at Amsterdam, and had found that Hudde was in possession of even more than Sluse

He corresponded with the mathematicians and intellectuals of the day; his correspondents included Blaise Pascal, Christiaan Huygens, John Wallis, and Michelangelo Ricci. He was appointed Chancellor of Liege and Counsellor and Chancellor to Prince Maximilian-Henry of Bavaria. He was elected a Fellow of the Royal Society in 1674. *Wik

1842 George Thom (2 July 1842 in Forgue near Huntly, Scotland - 20 Dec 1916 in Aberdeen, Scotland) graduated from Aberdeen and then became Principal of Doveton College in Madras, India. He returned to Scotland as Vice-Principal of Chanonry School Aberdeen and then became Rector of Dollar Institution (later to become Dollar Academy). He held this post for 24 years. He was a founder member of the EMS and became the fifth President in 1886. *SAU

1847 Andrew Gray graduated from Glasgow University and was appointed assistant and secretary to Lord Kelvin. He became Professor of Physics at University College Bangor and then returned to Glasgow as Kelvin's successor. He produced many books and papers in both mathematics and physics.*SAU

His major scientific publications included works on electromagnetism, dynamics and Bessel functions. He also wrote a treatise on gyrostats.

1852 William Burnside (2 July 1852 – 21 August 1927) , whose Theory of Groups (1897, 1911) is now a classic. His suspicion that every group of odd order is solvable was proved in 1962 by Walter Feit and John G. Thompson. *VFR He is known mostly as an early contributor to the theory of finite groups. In 1897 Burnside's classic work Theory of Groups of Finite Order was published. The second edition (published in 1911) was for many decades the standard work in the field. A major difference between the editions was the inclusion of character theory in the second.

Burnside is also remembered for the formulation of Burnside's problem (which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it) and for Burnside's lemma (a formula relating the number of orbits of a permutation group acting on a set with the number of fixed points of each of its elements) though the latter had been discovered earlier and independently by Frobenius and Cauchy.

1862 Sir William Henry Bragg (2 July 1862 – 10 March 1942) was a pioneer British scientist in solid-state physics who was a joint winner (with his son Sir Lawrence Bragg) of the Nobel Prize for Physics in 1915 for research on the determination of crystal structures. During the WW I, Bragg was put in charge of research on the detection and measurement of underwater sounds in connection with the location of submarines. He also constructed an X-ray spectrometer for measuring the wavelengths of X-rays. In the 1920s, while director of the Royal Institution in London, he initiated X-ray diffraction studies of organic molecules. Bragg was knighted in 1920. *TIS

1876 Harriet Brooks, born July 2, 1876 in Exeter, Ontario, enjoyed the distinction of being the first graduate student to work with Ernest Rutherford, a giant (both physically and intellectually) of early atomic physics. They enjoyed a happy, productive period of collaboration until their lives diverged in dramatically different directions.

Harriet Brooks was the third of nine children born to Elizabeth Worden and George Brooks, a commercial traveler for a flour company. The family’s move to Montreal in 1894 proved fortunate for Harriet, who attended McGill University on scholarships and graduated with honors in mathematics and natural philosophy in 1898. That same summer, Rutherford arrived at McGill as a 28-year-old physics professor fired up about radioactivity.

Together, Brooks and Rutherford studied what he called “radium emanation.” Their joint paper, published in 1901 in the Transactions of the Royal Society of Canada, identified this mysterious substance as a heavier-than-air gas.

The new gas appeared to be another new radioactive element, though they dared not label it as such. At the time, no respectable scientist would boast of turning one element into another – a claim that smacked of alchemy. As the pace of discovery and understanding accelerated, however, “emanation” indeed proved to be a new addition to the periodic table: the element radon.

In pursuit of a doctoral degree (not then offered by McGill), Harriet Brooks continued her research as a Fellow in Physics at Bryn Mawr College in Pennsylvania. Again she distinguished herself, winning the Bryn Mawr President’s Fellowship for graduate study in Europe. Rutherford intervened to place her with his own mentor, J. J. Thomson at the Cavendish, where she spent the 1902-1903 academic year. Then, instead of returning to Bryn Mawr to complete her studies, she returned to McGill, to Rutherford. Here she made a startling discovery that she reported in a letter to Nature in 1904: In addition to releasing a gas, radium also ejected radioactive atoms that could accumulate on a non-radioactive surface.

This phenomenon, now known as radioactive recoil, was reported with excitement four years later by Lise Meitner and Otto Hahn. Rutherford told them right away that Harriet Brooks had seen the same thing well beforehand, and Hahn eventually credited her as the first observer when he wrote his autobiography.

Most likely following her heart, Harriet Brooks left McGill in 1905 to teach physics at Barnard College, the women’s part of Columbia University, where she was reunited with Bergen Davis, a fellow physicist she’d met at the Cavendish. In the summer of 1906, when she informed officials at the college of her engagement to Davis, they requested her resignation.

She stood up to the dean, claiming “a woman has a right to the practice of her profession and cannot be condemned to abandon it merely because she marries.” That said, she broke up with Davis and spent the following year as an independent researcher at the Curie lab in Paris.

Marie Curie had assumed directorship of the lab at the Sorbonne following her husband’s death in April 1906. She was pleased with Brooks, her first hire, and invited the talented young scientist to stay on for at least another year. Brooks chose instead to rejoin Rutherford, who had moved to the University of Manchester. Eager to welcome her again, Rutherford supported Brooks’s fellowship application with a sterling letter of recommendation, in which he insisted that “next to Mme. Curie she is the most prominent woman physicist in the department of radioactivity.”

Midway through these arrangements, marriage to an old flame from McGill took Harriet Brooks back to Montreal. As wife of physics instructor Frank Pitcher and mother of three children, she pursued no further study of radioactivity, though she helped other female researchers win scholarships through her involvement with the Canadian Federation of University Women. The Pitchers lost their son Charles to meningitis at age fourteen. They were stricken again when their eighteen-year-old daughter, Barbara, went missing between classes at McGill in March 1929 and was found weeks later, drowned.

Harriet Brooks died on April 17, 1933, after a lingering but undisclosed illness. She was 56 years old. Rutherford submitted a formal obituary notice to Nature describing her important contributions. He expressed his personal loss in a letter to a colleague:

“She was a woman of great personal charm as well as of marked intellectual interests. I am afraid her domestic life was not without serious trials which she bore with astonishing fortitude. My wife and I held her in great affection and her premature death is a grievous blow to us.”

*Linda Hall Library Org

Ernest Rutherford’s research group in Montreal, 1899. Harriet Brooks is at center rear; Rutherford is at far right (aip.org)

1885 Émile Henriot (2 July 1885 - 1 February 1961) was a French chemist notable for being the first to show definitely that potassium and rubidium are naturally radioactive.
He investigated methods to generate extremely high angular velocities, and found that suitably placed air-jets can be used to spin tops at very high speeds - this technique was later used to construct ultracentrifuges.
He was a pioneer in the study of the electron microscope. He also studied birefringence and molecular vibrations.
He obtained his DSc in physics in 1912 the Sorbonne, Paris, under Marie Curie. *Wik

1988 Stanisława Nikodym (née Liliental; 2 July 1897 — 25 March 1988) was a Polish mathematician and artist. She is known for her results in continuum theory, especially on Jordanian continuums.

While on leave from university in 1918–1919, Stanisława taught mathematics to soldiers in the Polish army.

Her doctoral thesis was titled On disconnecting the plane by connected sets and continua. She published three books and several articles before World War II broke out.

Among her findings were necessary and sufficient conditions for a subcontinuum of a Jordanian continuum to be Jordanian. She also established that if the intersection and union of two closed sets are Jordanian continua, then so are the sets themselves.

In the 1940s, she taught mathematics at Kenyon College in Gambier, Ohio, where her husband was also a member of the faculty.

After her husband's death in 1974, she donated their papers and her paintings to the Briscoe Center for American History at the University of Texas, Austin. Stanisława Nikodym died in Warsaw in 1988.

1925 Olga Arsenevna Oleinik (2 July 1925, 13 Oct 2001) Oleinik wrote over 370 published papers and eight books. Her main research was concerned with algebraic geometry, partial differential equations, and mathematical physics. Winner of numerous prizes including the 1952 Chebotarev Prize for her research on elliptic equations with a small parameter in the highest derivative, the 1964 Lomonosov Prize for research on asymptotic properties of the solutions of problems of mathematical physics, and the 1988 State Prize for her series of papers on the investigation of boundary-value problems for differential operators and theirs applications in mathematical physics. In 1985 she was awarded the honorary title of Honored Scientist of the Russian Federation for her achievements in research and teaching, and in 1995 was awarded the Order of Honor by the president of the Russian Federation. She was also the 1996 AWM Noether Lecturer.*Agnes Scott College,

1906 Hans Bethe, (July 2, 1906 – March 6, 2005) German-born American theoretical physicist who helped to shape classical physics into quantum physics and increased the understanding of the atomic processes responsible for the properties of matter and of the forces governing the structures of atomic nuclei. Bethe did work relating to armour penetration and the theory of shock waves of a projectile moving through air. He studied nuclear reactions and reaction cross sections (1935-38). In 1943, Oppenheimer asked Bethe to be the head of the Theoretical Division at Los Alamos on the Manhattan Project. After returning to Cornell University in 1946, Bethe became a leader promoting the social responsibility of science. He received the Nobel Prize for Physics (1967) for his work on the production of energy in stars. *TIS

Bethe in 1967 *wik

1926 Rebeca Cherep de Guber (2 July 1926 – 25 August 2020) was an Argentine mathematician, university professor, textbook author and 1960s pioneer in the development of computer science in Argentina.

She completed her undergraduate studies at the National University of La Plata, earned her PhD in mathematics, and taught at the Faculties of Exact and Natural Sciences and Engineering at the University of Buenos Aires.

She married José Guber, an engineer, and they had at least one child, Rosana Guber.

In 1960 she was part of the group of scientists and teachers who created the Argentine Calculation Society, under the direction of Manuel Sadosky, with whom, years before, she had written the textbook, Elements of Differential and Integral Calculus. In the years since its first publication, the text has been widely disseminated among advanced students of science and engineering, and republished many times.

The Calculation Institute (IC) of the Faculty of Exact and Natural Sciences was created around 1959. Rebeca Guber took over as Technical Secretary on June 6, 1960. A few months later, the computer named Clementina (which was installed in 18 metal cabinets stretching 18 metres (59 ft) long) became known as the first computer installed for scientific research in Argentina and began its operations at the IC.

After the beginning of the coup (1966) Rebeca Guber, Juan Ángel Chamero and David Jacovkis resigned their positions and under the leadership of Manuel Sadosky, they founded a consultancy firm called Scientific Technical Advisors (ACT), in part to prevent the institute's lines of research and work from being totally abandoned.

After the coup ended in 1983, Guber continued to work with Sadosky when he was named the Nation's Secretariat of Science and Technology.

Guber died in 2020 from COVID-19.

DEATHS
1566 Nostradamus, French astrologer died on this day (b. 1503). I wonder if he predicted THIS in his prophacies.

1591 Vincenzo Galilei ( c. 1520 – 2 July 1591) Italian, music theorist, lutenist and composer, who as the father of Galileo Galilei, adopted experimentation to prove aspects of acoustics, and may thus have influenced his son, Galileo, away from pure, abstract mathematics and towards making experiments and investigation. Vincenzo's discoveries in acoustics included some of the physics of vibrating strings and columns of air. In particular he was the first to show that the ratio of an interval was proportional to string lengths but varied as the square of the tension applied to the strings and as the cubes of volumes of air. He recognized the superiority of equal tempered tuning and compiled a codex of pieces illustrating the use of all 24 major and minor keys as early as 1584.*TIS

1613 Bartholomeo Pitiscus (August 24, 1561 – July 2, 1613) was a Polish theologian who first coined the word Trigonometry. *SAU Pitiscus achieved fame with his influential work written in Latin, called Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus (1595, first edition printed in Heidelberg), which introduced the word "trigonometry" to the English and French languages, translations of which had appeared in 1614 and 1619, respectively. It consists of five books on plane and spherical trigonometry. Pitiscus is sometimes credited with inventing the decimal point, the symbol separating integers from decimal fractions, which appears in his trigonometrical tables and was subsequently accepted by John Napier in his logarithmic papers (1614 and 1619).*Wik

Front Cover of the 1612 edition of Trigonometriæ
sive de dimensione triangulorum libri quinque.

1621 Thomas Harriot (Oxford, c. 1560 – London, 2 July 1621) died of a cancerous ulcer on his left nostril. While in America in 1586 he learned to “drink” tobacco smoke from the Indians. This probably makes him the earliest recorded tobacco fatality. He is best known for his contributions to algebra, including the invention of the symbol for less than, $ \lt $ and greater than, $ \gt $ . He might have adopted this symbol from a decoration on an Indian’s back. See C. L. Smith, “On the origin of ‘<’ and ‘>’,” The Mathematics Teacher, 57(1964), 479–481 for a picture of this Indian.*VFR
He also is credited with the mathematical symbol for "therefore" $\therefore $ His executors posthumously published his Artis Analyticae Praxis on algebra in 1631; Nathaniel Torporley was the intended executor of Harriot's wishes, but Walter Warner in the end pulled the book into shape. It may be a compendium of some of his works but does not represent all that he left unpublished (more than 400 sheets of annotated writing). It isn't directed in a way that follows the manuscripts and it fails to give the full significance of Harriot's writings.*Wik He introduced a simplified notation for algebra and his fundamental research on the theory of equations was far ahead of its time. He was able to solve equations, even with negative or complex roots. However, he published no mathematical work in his lifetime. (Artes analyticae praxis, posthumous, 1631). Especially early in his career, he worked on navigation for his patron Walter Raleigh. Harriot carried out extensive telescopic observations of the satellites of Jupiter and of sunspots. When investigating optics, he discovered the sine law and measured the refractive indices of 13 different substances. He investigated free motion and motion resisted in air, and ballistic curves.*TIS Thomas Harriot was an English mathematician who did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look almost like a present day solution.*SAU
The solution of quadratic equations by the method of factoring was often referred to as Harriot's method because of his introduction of the method in his writing.
It is not possible today to find Harriot's grave. Although he was buried near the altar of St Christopher le Stocks in London,the church was destroyed in the great fire of 1666. There is a plaque in the entrance hall of the Bank of England, which is close to the site of Harriot's grave. It reproduces the original Latin wording of his epitaph.(p474) An English translation would read:

Stay, traveler, lightly tread;
Near this spot lies all that was mortal
Of that most celebrated man Thomas Harriot.
He was that most learned Harriot . . .
Who cultivated all the sciences and excelled in all . . .
A most studious searcher after truth . . .

1644 William Gascoigne (1612 – 2 July 1644) was an English astronomer, mathematician and maker of scientific instruments from Middleton, Leeds who invented the micrometer. He was one of "nos Keplari" a group of astronomers in the north of England who followed the astronomy of Johannes Kepler which included, Jeremiah Horrocks and William Crabtree. Gascoigne's micrometer is shown at right from a drawing by Hooke. Gascoigne, was working on a Keplerian optical arrangement when a thread from a spider’s web happened to become caught at exactly the combined optical focal points of the two lenses. When he looked through the arrangement Gascoigne saw the web bright and sharp within the field of view. He realized that he could more accurately point the telescope using the line as a guide, and went on to invent the telescopic sight by placing crossed wires at the focal point to define the centre of the field of view. Gascoigne died at the Battle of Marston Moor, Yorkshire, *Wik

1874 Gouverneur Emerson (August 4, 1795 – July 2, 1874) American physician, statistician and agriculturalist who prepared a series of tables of deaths and causes in Philadelphia, during thirty years from 1807. These showed, for example, the excessive mortality of males during childhood. He began practice in Philadelphia on 4 Aug 1820, where yellow fever broke out a few weeks later, with 73 deaths by that fall. Emerson recorded cases, dates, locations, and outcomes. He concluded no current medical treatments was especially effective. When smallpox reappeared there, with 325 deaths in 1824, Emerson drafted a bill for control measures. There were only 6 cases of smallpox in the city in 1825, and 3 in 1826. In retirement, he turned to peach culture, and studied phosphate and guano fertilizers. *TIS

1933 Harriet Brooks (July 2, 1876 – April 17, 1933) was the first Canadian female nuclear physicist. She enjoyed the distinction of being the first graduate student to work with Ernest Rutherford, a giant (both physically and intellectually) of early atomic physics. They enjoyed a happy, productive period of collaboration until their lives diverged in dramatically different directions.

Harriet attended McGill University on scholarships and graduated with honors in mathematics and natural philosophy in 1898. That same summer, Rutherford arrived at McGill as a 28-year-old physics professor fired up about radioactivity.

Harriet Brooks died on April 17, 1933, after a lingering but undisclosed illness. *Linda Hall Library Org

Ernest Rutherford’s research group in Montreal, 1899. Harriet Brooks is at center rear; Rutherford is at far right (aip org)

1947 Nikolai Grigorievich Chebotaryov (15 June [O.S. 3 June] 1894 – 2 July 1947) proved his density theorem generalising Dirichlet's theorem on primes in an arithmetical progression*SAU

1963 Seth Barnes Nicholson (November 12, 1891 – July 2, 1963) was an American astronomer best known for discovering four satellites of Jupiter. As a graduate student at the University of California, while photographing the recently- discovered 8th moon of Jupiter with the 36-inch Crossley reflector, he discovered a 9th (1914). During his life career at Mt.Wilson Observatory, he discovered two more Jovian satellites (1938) and the 12th (1951), as well as a Trojan asteroid, and computed orbits of several comets and of Pluto. His main assignment at Mt. Wilson was observing the sun with the 150-foot solar tower telescope, and he produced annual reports on sunspot activity and magnetism for decades. With Edison Pettit, he measured the temperatures of the moon, planets, sunspots, and stars in the early 1920s. *TIS

Nicholson at his spectroscope

2002 Daniel Chonghan Hong (3 Mar 1956; 2 Jul 2002 at age 46) Korean theoretical physicist specializing in statistical physics and nonlinear dynamic physics, who with colleague Hugo Caram, originated the void diffusing-void model of granular flow, which is recognized as an effective theoretical treatment for a broad range of dynamical phenomena in granular media. In general, his work ranged from percolation network, viscous fingering, granular flows to traffic equations. He studied and taught in America from 1981, and wrote articles for popular magazines on various topics. He died at the young age of 46 of cardiac arrest. *TIS

2016 Rudolf Emil Kálmán (May 19, 1930 – July 2, 2016) is a Hungarian-American electrical engineer, mathematical system theorist, and college professor, who was educated in the United States, and has done most of his work there. He is currently a retired professor from three different institutes of technology and universities. He is most noted for his co-invention and development of the Kalman filter, a mathematical formulation that is widely used in control systems, avionics, and outer space manned and unmanned vehicles. For this work, U.S. President Barack Obama awarded Kálmán with the National Medal of Science on October 7, 2009. *Wik

2017 Marjorie Ruth Rice (née Jeuck;Feb 16, 1923–july 2, 2017) was an American amateur mathematician most famous for her discoveries of pentagonal tilings in geometry.

Rice was born in St. Petersburg, Florida.

Marjorie Rice was a San Diego mother of five, who had become an ardent follower of Martin Gardner's long-running column, "Mathematical Games", which appeared monthly, 1957–1986, in the pages of Scientific American magazine. By the 1970s, Gardner was a popular science writer and amateur mathematician. Rice said later that she would rush to grab each issue from the mail before anyone else could get it, especially her son who subscribed to the magazine.

In 1975, Rice read Gardner's July column, "On Tessellating the Plane with Convex Polygon Tiles", that discussed what kinds of convex polygons can fit together perfectly without any overlaps or gaps to fill the plane. In his column, Gardner indicated that "the task of finding all convex polygons that tile the plane …. was not completed until 1967 when Richard Brandon Kershner … found three pentagonal tilers that had been missed by all predecessors who had worked on the problem". Gardner was repeating Kershner's claim that the list of convex pentagon tilers was complete. But within a month, Gardner received an example, by one of his readers, Richard James III, of a new convex pentagon tiler, and published this news in his December 1975 column.

Inspired by this new discovery, Rice decided to try to find other new pentagon tilers. Despite having only a high-school education, but a keen interest in art, she began devoting her free time to discovering new pentagonal tilings, ways to tile a plane using pentagons. She worked on the problem in her free time and through the 1975 holiday season "by drawing diagrams on the kitchen table when no one was around and hiding them when her husband and children came home, or when friends stopped by". She even developed her own system of notation to represent the constraints on and relationships between the sides and angles of the pentagons.

By February 1976, she had discovered a new pentagon type and its variations in shape and drew up several tessellations by these pentagon tiles. She mailed her discoveries to Gardner using her own home-made notation. He, in turn, sent Rice's work to Doris Schattschneider, an expert in tiling patterns, who was skeptical at first, saying that Rice's idiosyncratic notation system seemed odd, like "hieroglyphics". But with careful examination, she was able to validate Rice's results.

By October 1976, Rice had discovered 58 pentagon tilings that needed two pentagons stuck together in order to tile "transitively" (most of them previously unknown), which she arranged into 12 classes By December 1976, she had discovered two additional new types of tessellating pentagons and over 75 distinct tessellations by pentagons that were in blocks that could be seen as "double hexagons". In December 1977, she made her fourth discovery of a new type of pentagon tile and by then had enumerated 103 "2-block transitive" pentagon tilings.

Rice had completed half of a correspondence course in commercial art before she married. Throughout her investigations, she explored how to use pentagonal tilings as grids on which to overlay tessellations of flowers, shells, butterflies and bees.

Rice's discoveries were never published in Gardner's Scientific American columns, but were revealed in an addendum to his original column that was included in his 1988 collection of columns, where he declared her discoveries "fantastic achievements".

Gardner sent Rice's work to Doris Schattschneider, who was an expert in tiling patterns. Schattschneider was skeptical at first, but upon careful examination, was able to validate Rice's results. Schattschneider not only helped Martin Gardner publicize the pentagon tilings discoveries of Rice, but lauded her work as a significant discovery by an amateur mathematician.

In 1995, at a regional meeting of the Mathematical Association of America held in Los Angeles, Schattschneider convinced Rice and her husband to attend her lecture on Rice's work. At the conclusion of the talk, Schattschneider introduced the amateur mathematician who had advanced the study of tessellation. "And everybody in the room . . . gave her a standing ovation."

Much of Rice’s investigations remain unpublished, in that only the product of her investigations are shown. How she devised these is not generally shown. However, some of her investigations are indeed shown in The Mathematical Gardner, a compilation of articles in honor of the late Martin Gardner, with Doris Schattschneider’s article In Praise of Amateurs (mostly concerning background detail on Rice’s pentagon tiling findings), pages 140-166. Pages 154-155 contain numerous convex pentagon tilings"

Four of Rice's pentagon tilings *Wik

Doris Schattschneider *SAU

2021 Dorothy Mary Elizabeth Foster (3 December 1933 Darlington, County Durham, England , 2 July 2021 St Andrews, Fife, Scotland)

Dorothy Foster studied at Bedford College and was awarded a Ph.D. by the University of London in 1960. Except for a few years as an Assistant Lecturer in Mathematics at Royal Holloway College, London, she spent her whole career at the University of St Andrews. She was an expert on geometric number theory.

She died "... peacefully, in her sleep, in St Andrews, aged 87 years, on 2nd July, 2021." *SAU

*WM = Women of Mathematics, Grinstein & Campbell