Friday, 6 March 2026

Infinite Radical Sequences, Still He Persisted.

  

I  hope women of the world can forgive my usurpation of the phrase from the Women's Movement, but the idea applies as I return again to the topic of infinite radicals.

It is said that Ramanujan posed the above problem to the Journal of Indian Mathematical Society:in 1911.  I use it on my "On This Day in Math" blog for a number fact on January third since it is the third day of the year.  Because the problems of analysis from infinite series often dances at the edge (or outside) my understanding of pure mathematics, I always question my assumptions about them, and so for several years I have asked about a seeming extension (or perhaps contraction) of this infinite sequence.  What happens when we chop off one layer from the front.       My thinking went like this:

If we take the expression and square both sides we get \( 9= 1+2 \sqrt{1+3 \sqrt{1+...}} \)

And doing the obvious arithmetic to clear the preamble before the first radical we arrive at  \(4=  \sqrt{1+3 \sqrt{1+...}} \)

Now if we repeat the process of squaring and simplifying the result a second time we get \(5=  \sqrt{1+4 \sqrt{1+...}} \)   and thus, as they say, "to Infinity".   

Just to make it easier, I have included in the remainder of this post some earlier thoughts about different infinite nested radicals exploring them on my on... 

___________________ Reposted material from Dec, 2009 ________________________________

Recently (2009) someone on the Calculus EDG asked about the value of.  I sent a link to some work I had done a while ago exploring the same idea, and extending to finding the value of 
. I have picked out some parts below, but you can see the rest at this link. (apparently this link has been lost in the internet .  I tried the Wayback machine but it seems to be an incomplete copy.  Ir you are way more savvy than me, and who isn't really, then maybe you'll do better and share what you find.) This is a very old Word Document so give it some time to load. Hopefully it is worth while. Dave Renfro then sent me a copy of some papers about the topic, including this one from a 1935 American Mathematical Monthly.  

When you take the iterated square root of a number, such as \(x = \sqrt{n+ \sqrt{n+ ...}} \) and then square both sides, you get \(x^2 = x + n\).  This means that we can find solutions using basic quadratic solution approaches, and then find solutions that produce integer values of x.  The positive solution becomes \( \frac{\sqrt{4n+1}+1}{2} \) 

One of the nice things I discovered was that the iterated square roots of 2 was not the only number that gave an integer answer.  In fact, 2, 6, 12, 20, 30.... all were equal to integer values... This sequence is the pronic or oblong numbers, which are twice the triangular numbers. These numbers can be expressed as (n)(n+1) .  It took me a moment to realize why they are the ones that would work. These are numbers that, when multiplied by four and increased by 1, become perfect squares, \( 4 (n^2+n)+ 1 = 4n^2 + 4n+1 = (2n+1)^2 \).  And the square root, being an odd (2n+1) number so that when 1 is added, we get a number divisible by 2.  

It seems, according to the Herschfeld article,  that the problem was a common topic in the Columbia classes of Dr. Edward Kasner.  Kasner, of course, is known for his part in the creation of the term "googol" for 10^100.  If your interested in any of these topics, check either or both the links above .   


I had not yet tried to consider the roots of the cube root of (a+cube root(a+ .... etc)) and so I wanted to take a shot.. By the same process I had used before, the value would be the solution to x^3-x-n=0 .  

If the iterated value was 1, the value approaches about x=~1.32472.  For n=2 the value is x=~1.52138.  By the time we get to n=6, we get x=2.  The actual solution for any n is

OK... that really isn't very much fun to play with, but after some experimenting, I came up with the fact that the following sequence of numbers produced integer values when iterated; 6, 24, 120,  210... ; or perhaps it is more revealing to write them a different way (1*2*3) , (2*3*4), (3*4*5)... so they were sort of the three dimensional pronic numbers, the products of three consecutive integers.  (I have never seen a name for these, so I'm introducing hexonic, because they are all divisible by six.  Sphenic is also appropriate since it is the Greek root for wedge shaped, but it seems overused for any number with three distinct factors.)
I could not manipulate the above equation to make it clear that these were the only values as I had with the quadratic, but it got me thinking, what if I did fourth roots ?   (This is the point where a more clever mathematician would have said hmmmm, squares are solved by x2 - x -n=0; and and cubes by  x3 - x -n=0, maybe there is a pattern) 

Extending the solutions for square and cube roots, I tried 1*2*3*4 = 24.... but the solution of \(n^4 - n- 24=0\) was NOT 2; in fact, it was about 2.1617???   (Yep, guess who picked the wrong pattern to pursue? 

Exploring I found that n=2 was a solution to \(n^4 - n- 14=0\).  And n=3 was a solution when the constant was 78. The sequence is 14, 78, 252, 620, 1290,...  These values follow the form n*(n-1)*(n^2+n+1)
I realized, somewhat belatedly, that you could generate these sequences by simply using nk - n ( \(2^4-2=14, 3^4-3 = 78, etc \)for integer values of k, and factoring the same would give you the simplified form of the expression.  And it seemed true for all the others.  The pronic numbers 2, 6, 12 are \(2^2-2, 3^2-3, 4^2-4\)  and I'll let you convince yourself that the cubes root iterations works the same.


After struggling with solving n^3 - n -k=0 I realized that I could just start with values of n, and find out what k came out to be. A very late "aha" moment.  So for n=2, 23 - 2 = ??? and six pops out like Alg I.  
 

The fourth powers no longer followed the pronic, hexonic, and whatever name I would have given n(n=1)(n+2)(n+3).  But they do seem to follow a pattern of a pronic times number of the form (n^2 + n +1)  And a little algebra factoring n^4-n will show why.

There is a familiar quotation about forests and trees that seems to apply here, but it came to me somewhat late. 


But sometimes, that's how my mind works... do it the hard way first.

On This Day in Math - March 6

 

Chapel of the Holy Shroud, Santa Sindone


Nobody since Newton has been able to use geometrical methods to the same extent for the like purposes; and as we read the Principia we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them we marvel what manner of man he was who could use as a weapon what we can scarcely lift as a burden.
~William Whewell

The 65th day of the year; 65 is the smallest hypotenuse of two different primitive Pythagorean triangles (and of two other triangles that are not primitive) with all integral sides. (Don't just sit there, find them!)
John Golden@mathhombre not only found them, he made the image below.  

And \( 65 = 1^5 + 2^4 + 3^3 + 4^2 + 5^1 \) *jim wilder ‏@wilderlab

OR, \(65= 0^2 + 1^4 + 2^5 + 3^3 + 4^1 + 5^0 \) *@Expert_says


65 is the constant of a 5x5 normal magic square.
A magic square with the integers 1 through 25 has a sum of 65 in each row, column, and major diagonal.

Euler found 65 integers, which he called "numeri idonei," that could be used to prove the primality of certain numbers.[idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as \(x^2 ± Dy^2\) (where x2 is relatively prime to Dy2) is a prime, prime power, twice one of these, or a power of 2. In particular, a number that has two distinct representations as a sum of two squares (such as 65) is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.]

65 is the difference of fourth powers of two consecutive  primes. And a note about fourth powers of primes.  For any prime greater than five, the last digits of a p^4 either ends in an odd digit followed by six, or an even digit followed by one.



EVENTS

1619  Edmund Gunter appointed Gresham Professor of Astronomy. In 1619 the wealthy but earnest Sir Henry Savile put up money to fund Oxford University's first two science faculties, the chairs of astronomy and geometry. Gunter applied to become professor of geometry but Savile was famous for distrusting clever people, and Gunter's behavior annoyed him intensely. As was his habit, Gunter arrived with his sector and quadrant, and began demonstrating how they could be used to calculate the position of stars or the distance of churches, until Savile could stand it no longer. "Doe you call this reading of Geometric?" he burst out. "This is mere showing of tricks, man!" and, according to a contemporary account, "dismissed him with scorne." He was shortly thereafter championed by the far wealthier Earl of Bridgewater, who saw to it that on 6 March 1619 Gunter was appointed professor of astronomy in Gresham College, London. (Henry Briggs, who received the position of Gresham Professor of Geometry when Gunter was passed over also supported Gunter, and nominated him for the Astronomy position.) This post he held till his death. Gunter created the first logarithmic scale. Gunter's scale or Gunter's rule, generally called the "Gunter" by seamen, is a large plane scale, usually 2 feet (0.61 m) long by about 1½ inches broad (600 mm by 40 mm), and engraved with various scales, or lines. On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, etc.), and on the other side the corresponding artificial or logarithmic ones. By means of this instrument questions in navigation, trigonometry, etc., are solved with the aid of a pair of compasses. It is a predecessor of the slide rule, a calculating aid used from the 17th century until the 1970s.
He is also known for Gunter's chain , a geodetic measuring device used for land survey. When the Northwest territory (Ohio, Indiana, Michigan, Illinois etc) was created, the decreed official measure was the Gunther Chain.*Wik On a visit to Stratford on Avon while at Hall's croft, the home of Shakespeare's daughter Susanna and her husband, Dr John Hall, I came across an early map of the town and the only legend shown was in Gunter's Chains. Watching an English Cricket match one day in Dec of 2006, I realized that the length of the bowling area (between the two wickets) is one chain also.  (I have no record of where I got the image above, or if it is part of a larger image.  If you have info. please share.)

And the measure of an acre??? Perhaps you've wondered how they settled on 43,560 sq ft for an acre of land.  Not nan easy to remember number, but maybe if we work backwards... dividing by 9 we get the number of square yards in an acre, 4840. Ok, that's obviously ten of something,  10 x 484 square yards.  Then if we notice that 484 is a perfect square, 22x22 or one square chain.  So the simple way to remember the area of an acre is 10 square chains.  Oh just two more, and acre is four square roods, so a rood is 1/4 of an acre, and a perch is 1/16 of a square chain, or 1/160 of an acre (30.25 sq yards). Aren't you glad we avoided all that Metric stuff in the USA?
OOOH, and a good teacher joke, Tell your young charges that it was King Henry the First who decreed that the yard should be the distance from his nose to the tip of his outstretched finger..... (wait for it)... pause..."and we still refer to him today as a ruler of England!  (Ignore the moaning.)


In 1646,Joseph Jenckes Sr. also spelled Jencks and Jenks, was a bladesmith, blacksmith, mechanic, and inventor who was instrumental in establishing the Saugus Iron Works in Massachusetts Bay Colony where he was granted the first machine patent in North America from the General Court of Massachusetts.
He received a 14-year patent for a new kind of water-driven machine to make scythes, sawmill saw blades, and other edged tools.
A master mechanic, an operator of an extensive foundry and metal works, and an expert blacksmith; Established first iron & steel works in Lynn, Massachusetts.  
You may sometimes see this mis-written as first patent in US.  In 1641 the Massachusetts General Court gave Samuel Winslow an exclusive right to utilize a new process of making salt for 10 years. The case is unofficially known of as the first "patent" in America.  
Jenckes was raised in a family of London cutlers and found employment west of London at a sword factory. After his wife and daughter died, and about the time the sword factory closed, he left his only surviving child with family and immigrated to New England.
The son he left behind in England, Joseph Jenckes Jr., joined him at Saugus and later founded the town of Pawtucket in the Colony of Rhode Island and Providence Plantations. Other notable descendants include a co-founder of Brown University and a governor of colonial Rhode Island.


In 1661, the Royal Society, London, England, elected Sir Robert Moray as their first president. *TIS

1665 first appearance of the Philosophical Transactions of the Royal Society. The Journal des sçavans (later renamed Journal des savants), founded by Denis de Sallo, was the earliest academic journal published in Europe, that from the beginning also carried a proportion of material that would not now be considered scientific. The first edition appeared as a twelve page quarto pamphlet on Monday, 5 January 1665. This was shortly before the first appearance of the Philosophical Transactions of the Royal Society, on 6 March 1665. *Wik

1689 Edmond Halley first wrote about diving equipment in a paper of 6 March 1689, perhaps prompted by his work on the Thames survey undertaken around that time. Halley proposed a mobile diving bell built on four wheels, and while he didn’t build that particular bell, he did build another as part of his salvage work on the wreck of the Guynie frigate. *halleyslog




*http://laurenroyal.com/
1703 Robert Hooke is buried at the church of St Helen, Bishopsgate, London. He had died on March 3. The only known portrait of Robert Hooke, which hung in Gresham College, mysteriously disappeared shortly after his death. A memorial window to him was destroyed by a bomb in 1992.
Hooke was elected to the Royal Society in 1663 and became its curator for the rest of his life. He was Professor of Geometry at Gresham College, London, and lived there as a bachelor until his death in 1703.
For those who do not know his story, Lisa Jardines, biography is wonderful. 




1741 Euler writes to Goldbach that he has proved “a theorem of Fermat’s” according to which primes p = 4n + 3 cannot divide a sum of two squares \( a^2 + b^2 \) except when both a and b are divisible by p. Correspondence of Euler and Goldbach.

1766 d’Alembert writes Lagrange to tell him Euler is leaving Berlin Academy:
Mr Euler is leaving, he says, for St.Petersburg because of some unhappiness he has had in Berlin. I wrote to him to dissuade him. If he leaves, and you want to replace him, you have only to write me and I will do my best to serve you.
Before 1766, Frederick II of Prussia had more than once invited both d’Alembert and Lagrange to move to Berlin. The d'Alembert had declined the offer and suggested the name of his Turinese friend. But Lagrange, even though he was on good terms with Euler, did not relish a "cohabitation" with him in the Berlin Academy. It seems he may have feared Euler would overshadow him. *Mauro Allengranza, Stack Exchange




1805 Legendre introduced least squares. Gauss had them ten years earlier but had not published, so some controversy ensued. *VFR It was on this day that he published the little 80 page appendix, Nouvelle me'thodes pur la determination des orbites des cometes. "Of all the principals that can be proposed for this purpose, I think there is none more general, more exact, or easier to apply,... it consists of making the sum of the squares of the errors a minimum." *Stephen M. Stigler, The History of Statistics.
Legendre appears to have discovered the method in early 1805, and Robert Adrain may have "discovered" it in Legendre's 1805 book (Stigler, 1977, 1978), but in 1809 Gauss had the temerity to claim that he had been using the method since 1795, and one of the most famous priority disputes in the history of science was off.





1815 Wilhelm Olbers, an amateur German astronomer who was a doctor by profession, discovered the periodic comet now named for him.  This amateur astronomer would discover many comets, and his calculating method would change the science.  He became a lifelong friend of Gauss after their correspondence regarding the discovery of Ceres in January of 1802.  He would allow Guass to name the planet (now, asteroid, a term not in use then) that he discovered in 1807, Vesta. *Wik
Sketch of 13P/Olbers on 14 October 1887 by William Robert Brooks



1832 Gauss responds to his “old, unforgettable friend,” Farkas (Wolfgang) Bolyai, that he has been working on non-Euclidean geometry “in part already for 30–35 years.” In the same letter Gauss points out several flaws in Euclid. *VFR Bolyai had included the work of Janos, his son, on non-Euclidean Geometry in a letter to Gauss on the 20th of June 1831.. and again on the 16th of January 1832 Farkas sent the Appendix to Gauss again with another letter in which he wrote: ``My son appreciates Your critique more than that of whole Europe and it is the only thing he is waiting for''. In his response, One of Gauss' well-known sentences was: ``if I praised your son's work I would praise myself''. The letter deeply afflicted and upset János Bolyai, although it reflects appreciation, too: ``... I am very glad that it is my old friend's son who so splendidly preceded me'' *Komal Journal
Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.




In 1869, Dmitry Mendeleev published his first version of the periodic table of the elements. He was a Russian chemist who developed the periodic classification of the elements. In his final version of the periodic table (1871) he left gaps, foretelling that they would be filled by elements not then known and predicting the properties of three of those elements. *TIS  Mendeleev had written the properties of elements on pieces of card and tradition has it that after organizing the cards while playing patience he suddenly realized that by arranging the element cards in order of increasing atomic weight that certain types of element regularly occurred.*Royal Society of Chemistry




1896 Dutch cryogenic physicist, Heike Kamerlingh Onnes, writes to James Dewar in England to explain the reason he had not made any recent experiments in cooling gases: "..you will be astonished to hear. The municipality of Leiden has made objections as to my working with condensed gases and has not been content with asking that additional means of precaution are taken, but is gone so far to claim in August last that my cryogenic laboratory be removed from the city! " *archive of the Kamerlingh Onnes Laboratory




 1896    Detroit Free Press reported:
"The first horseless carriage seen in this city was out on the streets last night. It is the invention of Charles B. King, a Detroiter, and its progress up down Woodward Avenue about 11 o’clock caused a deal of comment, people crowding around it so that its progress was impeded. The apparatus seemed to work all right, and went at the rate of five or six miles an hour at an even rate of speed."
King would later work at several start-up automakers and launch King Motor Cars in 1910 — becoming the first U.S. automaker to offer cars with the steering wheel on the left and the first affordable V-8. His company eventually became part of Studebaker.
*Yahoo



1899 "Aspirin" (acetylsalicylic acid) patented by Felix Hoffmann at German company Bayer.  Hoffmann was a young pharmacist working for the German pharmaceutical company Bayer. The trademark name is aspirin. Hoffmann, who was said to be seeking an effective pain reliever for his father's rheumatism, successfully synthesized acetylsalicylic acid in August 1897.
Hoffmann first claimed to be the "inventor" of aspirin (as opposed to just the synthesizer) in a footnote to a German encyclopedia published in 1934
In 1949, ex-Bayer employee Arthur Eichengrün published a paper in Pharmazie, in which he claimed to have planned and directed Hoffman's synthesis of aspirin along with the synthesis of several related compounds. He also claimed to be responsible for aspirin's initial surreptitious clinical testing. Finally, he claimed that Hoffmann's role was restricted to the initial lab synthesis using his (Eichengrün's) process and nothing more. Eichengrün died the same month he published in Pharmazie.





In 1913, this date was written by Niels Bohr on his first paper describing his new ideas on atomic structure, and mailed to his mentor, Ernest Rutherford. It was one of three historic papers he wrote on this subject. *TIS
In Bohr’s model, electrons moved around the atomic nucleus in circular orbits, but those orbits had set discrete energies, and electrons could gain or lose energy only by moving from one orbit to another, absorbing or emitting radiation as necessary. While it is still taught in introductory physics classes, the Bohr model is not quite correct. Nonetheless, this pioneering work earned Bohr the 1922 Nobel Prize in Physics, “for his services in the investigation of the structure of atoms and of the radiation emanating from them.” 
Bohr model of Carbon Atom




On this day in 1930, Kurt Gödel received his Ph.D. from the University of Vienna for a dissertation, directed by Hans Hahn, that showed the completeness of first order logic (every valid first order formula is provable).




*Thought Co

1950 
Silly Putty goes on sale in the US.  Though invented in 1943 by James Wright, Silly Putty was not a toy until Peter Hodgson packaged the goo in plastic eggs and sold them in 1950. 
In February 1950, Hodgson took Silly Putty to the International Toy Fair in New York, but most people there did not see the potential for the new toy. Luckily, Hodgson did manage to get Silly Putty stocked at both Nieman-Marcus and Doubleday bookstores.

A few months later, a reporter for The New Yorker stumbled across Silly Putty at a Doubleday bookstore and took home an egg. Fascinated, the writer wrote an article in the "Talk of the Town" section that appeared on August 26, 1950. Immediately, orders for Silly Putty started pouring in.




1953 James Watson and Francis Crick submitted to the journal Nature their first article on the structure of DNA. It was published in the 25 Apr 1953 issue. "We wish to put forward a radically different structure for the salt of deoxyribose nucleic acid. This structure has two helical chains each coiled around the same axis... Both chains follow right-handed helices... The novel feature of the structure is the manner in which the two chains are held together by purine and pyrimidine bases... They are joined together in pairs, a single base from one chain being hydrogen-bonded to a single base from the other chain, so that the two lie side by side with identical z-co-ordinates. One of the pair must be a purine and the other a pyrimidine in order for bonding to occur."*TIS

*The DNA Store




1967 A study of twelve industrial nations revealed that mathematics achievement is highest in Japan, lowest in the U.S. *VFR  
In 2017 Pew Research reported: Feb 15, 2017 — U.S. students continue to rank near the middle, and behind many other advanced industrial nations, in international math, science and reading

1986 USSR's Vega 1 flies by Halley's Comet at 8,889 km. Vega 1 encountered Comet Halley on March 6, 1986, and Vega 2 three days later. The flyby velocity was 77.7 km/s. Although the spacecraft could be targeted with a precision of 100 km, the position of the spacecraft relative to the comet nucleus was estimated to be known only to within a few thousand kilometers. 


*Space.com


1992 Michelangelo Virus Strikes: Concerns over the Michelangelo virus sparked a scare among everyone from personal computer users to world governments. As many as 5 million computers reportedly were at danger of contracting the virus, set to erase data on the March 6 anniversary of the artist's birth. In fact, Michelangelo spread to only a few thousand machines. *CHM


BIRTHS

1787  Joseph Ritter von Fraunhofer (6 March 1787 – 7 June 1826) was a German physicist and optical lens manufacturer. He made optical glass, an achromatic telescope, and objective lenses. He also invented the spectroscope and developed diffraction grating. In 1814, he discovered and studied the dark absorption lines in the spectrum of the sun now known as Fraunhofer lines. *Wik
The Great Dorpat Refractor built by Joseph Fraunhofer and completed in 1824 was the first modern, achromatic, refracting telescope. At the start of the 19th century, progress in astronomy was stifled by the lack of astronomical quality telescopes of sufficient aperture and manageability. There were long-focus, nonachromatic refractors, reflectors with speculum metal mirrors, and achromatic refractors of small aperture and mediocre design. Contributing to the construction and success of the Great Dorpat Refractor were P. L. Guinand's development of a process for making large disks of homogenous flint glass, Fraunhofer's improvement of the design and fabrication of the optical and mechanical components of the telescope, and F. G. W. Struve's skilled and dedicated use of the telescope and its accessories. The successors of the Great Dorpat Refractor were the giant refractors which were the mainstay of astronomy in the 19th century and which were not displaced until the early 20th century when the age of the giant reflectors began.*Astrophysics Data System

Illustration of solar spectrum drawn and colored by Joseph von Fraunhofer with dark lines named after him (1987 DBP's stamp on 200th anniversary of birthday of Fraunhofer):
Fraunhofer demonstrating the spectroscope:





1847 Johann Georg Hagen (6 Mar 1847, 5 Sep 1930) Austrian Jesuit priest and astronomer who made a catalog of variable stars (1890-1908). Working at the Vatican Observatory he reexamined for accuracy the listing of all of the NGC (New General Catalogue of Nebulae and Star Clusters) objects north of about -30 degrees. He published lists of errata in the NGC. During his observations, he observed dark nebulae, tenuous dark clusters of interstellar matter sometimes known as Hagen's clouds. These strange clouds have not been recorded by others, and are now attributed to optical illusions associated with visual observations. Jesuits have been involved in astronomy since 1551 when Fr. Christoph Clavius, SJ, a mathematician and astronomer helped Pope Gregory XIII reform the calendar.*TIS




1866 Ettore Bortolotti (6 March 1866 in Bologna, Kingdom of Sardinia (now Italy)
- 17 Feb 1947 in Bologna, Italy) Italian mathematician who worked in various areas in analysis. He was interested in the history of mathematics. .... recent work by Fowler has added much to our understanding of the concept of continued fractions as present in ancient Greek mathematics. Nevertheless, this work of Fowler does not diminish the value of Cataldi's contibutions.*SAU . He revealed the importance of Evangelista Torricelli’s infinitesimal results and vindicated Cataldi’s claim to the discovery of continued fractions. *VFR





1988 Annie Hutton Numbers (6 March 1897 in Edinburgh, Scotland - 10 April 1988 in High Wycombe, England) After a brief spell teaching she was appointed as Assistant Lecturer and Demonstrator at the Department of Chemistry at Edinburgh University. While on the staff of the University, Numbers undertook research towards the degree of Ph.D. which she took in 1926 for the thesis The influence of substituents on the optical rotatory power of compounds. She left her post at the Department after 1930 to become a teacher in Ipswich and then in High Wycombe, retiring in 1965. *SAU
Annie Hutton Numbers was a scientist, teacher and lifelong-learner. She graduated from the University of Edinburgh in 1918 with the degree of MA (Hons) in Mathematics and Natural Philosophy. In 1917 she joined the Edinburgh Mathematical Society, where she was a member for 16 years. *Wik

Who wouldn't want to have a math teacher whose name was Ms. Numbers?





1901 Naum Ilyich Akhiezer (6 March 1901 – 3 June 1980) was a Soviet mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators. He is also known as the author of classical books on various subjects in analysis, and for his work on the history of mathematics. He is the brother of the theoretical physicist Aleksander Akhiezer.*Wik





1937  Valentina Vladimirovna Tereshkova ( 6 March 1937- ) Soviet cosmonaut who was the first woman to fly in space, and is the only solo woman. She had worked in tyre and textile factories. She was selected (1961) as a cosmonaut for her expert skill in parachuting. She trained in a special woman-in-space program, and was the only one of the four women participants to complete a space mission. She was launched in Vostok 6 on 16 Jun 1963, two days after Valery F. Bykovsky in Vostok 5. Tereshkova made 48 orbits of Earth in 71 hours. The two cosmonauts landed on the same day, 19 Jun. Tereshkova left the program shortly after her return. She was honored with the title Hero of the Soviet Union. She went into space two decades before America's first woman astronaut, Sally Ride.





DEATHS

1683 Guarino Guarini (17 Jan 1624; 6 Mar 1683) Italian architect and theologian whose study of mathematics led him to a career in architecture in which he created the most fantastic geometric elaboration of all baroque churches. In his Santissima Sindone, Guarini created a diaphanous dome - a geometrical optical illusion in the dome made through the use of the actual structure which creates the illusion that the dome recedes farther up into space than it really does. He wrote two architectural treatises and other works that concentrate on his mathematical knowledge. Therein, Guarini discusses Desargue's projective geometry, which reveal a scientific basis for his daring structures. He worked primarily in Turin and Sicily, with his influence stretching into Germany, Austria and Bohemia.*TIS




1838  John Stevens (June 26, 1749 – March 6, 1838) American engineer and lawyer who invented the screw propeller (1802) (The screw propellor  and the multitubular boiler engine. His Phoenix was the first oceangoing steamboat (1809), and he operated the Juliana, the first steam-boat ferry. His interest in applying steam steam power to transportation began in the late 1780s. Stevens petitioned the U.S. Congress to establish a U.S. patent law (enacted 1790) and registered patents for his improved boiler and engine designs (1792). The sea voyage of the Phoenix paddle-wheel steamboat was from New York City to Philadelphia. On 11 Oct 1811, the Juliana, began operations as a ferry between New York, NY, and Hoboken, NJ. He demonstrated the first steam locomotive in the U.S. on his estate (1825)
Fig. 1 The Original Twin-Screw Engine and Boiler
of Col. John Stevens. Built in Hoboken, N.Y., in 1804.





1866 William Whewell (24 May 1794, 6 Mar 1866 at age 71) British scientist, best known for his survey of the scientific method and for creating scientific words. He founded mathematical crystallography and developed Mohr's classification of minerals. He created the words scientist and physicist by analogy with the word artist. They soon replaced the older term natural philosopher. (actually the use of scientist was a very slow process often not well received.(My blog about the long struggle of the word is here) Other useful words were coined to help his friends: biometry for Lubbock; Eocene, Miocene and Pliocene for Lyell; and for Faraday, anode, cathode, diamagnetic, paramagnetic, and ion (whence the sundry other particle names ending -ion). In meteorology, Whewell devised a self-recording anemometer. He was second only to Newton for work on tidal theory. He died as a result of being thrown from his horse. *TIS
In a single letter to Faraday on 25 April, 1834; he invented the terms cathode, anode and ion. The letter is on display at the Wren Library at Trinity College, Cambridge, UK.






1939 Carl Louis Ferdinand von Lindemann (12 Apr 1852, 6 Mar 1939 at age 86) He showed π transcendental not the root of any algebraic equation with rational coefficients), consequently the circle cannot be squared. (constructing a square with the same area as a given circle using ruler and compasses alone.) In 1873, Lindemann visited Hermite in Paris and discussed the methods which Hermite had used in his proof that e, the base of natural logarithms, is transcendental. Following this visit, Lindemann was able to extend Hermite's results to show that pi was also transcendental. *TIS(the image is of his tombstone.... note the square and circle with Pi inside.



1944 Aleksandr Petrovich Kotelnikov (20 Oct 1865 in Kazan, Russia - 6 March 1944 in Moscow, USSR) In 1927 he published one of his most important works, The Principle of Relativity and Lobachevsky's Geometry. He also worked on quaternions and applied them to mechanics and geometry. Among his other major pieces of work was to edit the Complete Works of two mathematicians, Lobachevsky and Zhukovsky. He received many honours for his work, being named Honoured Scientist in 1934, then one year before he died he was awarded the State Prize of the USSR. *SAU




2005 Hans Bethe (2 Jul 1906, 6 Mar 2005 at age 98), 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 armor 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



Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell 

Thursday, 5 March 2026

On This Day in Math - March 5

  


Terrestrial globe by Mercator dating from 1541. It is now in the museum collection of the Palazzo Ducale in Urbania, Italy, and is one of about 22 existing Mercator globes.*Wik


But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.
~Julian Lowell Coolidge


The 64th day of the year; 64 is the smallest power of two with no prime neighbor. (What is next value of 2n with no prime neighbor?)  Also the smallest even square number without a prime neighbor.

64 is also the smallest non-trivial positive integer that is both a perfect square and a perfect cube.

64 can be expressed as the sum of primes using the first four natural numbers once each, 41 + 23 = 64,  It can also be done with its reversal, 46 = 41 + 3 + 2.

There were 64 disks in Eduard Lucas' myth about the Towers of Hanoi.

 64 is also the number of hexagrams in the I Ching, and the number of sexual positions in the Kama Sutra. (I draw no conclusions about that information)

There are 64 ordered permutations of nonempty subsets of {1,..., 4}: Eighteenth-  and nineteenth-century combinatorialists call this the number of  (nonnull) "variations" of 4 distinct objects.

64 is a superperfect number—a number such that σ(σ(n)) = 2n. The sum of the divisors (including itself) of 64 is 127, and the sum of the divisors of 127, 1 and 127, add up to 128= 2*64. It is the last Year Day that is Super-Perfect.

And I was told that 64 is the maximum number of strokes used in a Kanji character.

Most mathematicians know the story of 1729, the taxicab number which Ramanujan recognized as a cube that was one more than the sum of two cubes, or the smallest number that could be expressed as the sum of two cubes in two different ways.  But not many know that 94 is part of the second such   \(64^3 + 94^3 = 103^3 + 1^3  \)   



EVENTS

In 1223 BC, the oldest recorded eclipse occurred, according to one plausible interpretation of a date inscribed on a clay tablet retrieved from the ancient city of Ugarit, Syria (as it is now). This date is favored by recent authors on the subject, although alternatively 3 May 1375 BC has also been proposed as plausible. Certainly by the 8th century BC, the Babylonians were keeping a systematic record of solar eclipses, and possibly by this time they may have been able to apply numerological rules to make fairly accurate predictions of the occurrence of solar eclipses. The first total solar eclipse reliably recorded by the Chinese occurred on 4 Jun 180 BC*TIS

 The Ugarit eclipse darkened the sky for 2 minutes and 7 seconds on May 3, 1375 B.C., according to an analysis of a clay tablet, discovered in 1948. Then, a report in the journal Nature in 1989 suggested, in fact, the eclipse actually occurred on March 5, 1223 B.C. That new date was based on an historical dating of the tablet as well as an analysis of the tablet’s text, which mentions the visibility of the planet Mars during the eclipse.

The Ugaritic texts are a corpus of ancient cuneiform texts discovered since 1928 in Ugarit (Ras Shamra) and Ras Ibn Hani in Syria, and written in Ugaritic, an otherwise unknown Northwest Semitic language. Approximately 1,500 texts and fragments have been found to date. The texts were written in the 13th and 12th centuries BC.

A tablet in the collection.



In 1590, Tycho Brahe discovered a comet in the constellation Pisces.*TIS    Prior to his death in 1601, he was assisted for a year by Johannes Kepler, who went on to use Tycho's data to develop his own three laws of planetary motion.



In 1616, Copernican theory was declared "false and erroneous" in a decree delivered by Cardinal Robert Bellarmine, and issued by the Catholic Church in Rome. Further, no person was to be permitted to hold or teach the theory that the earth revolves around the sun. When Galileo subsequently violated the decree, he was put on trial and held under house arrest for the final eight years of his life. *TIS Copernican theory was declared "false and erroneous" by the 11 theologians, appointed by the Pope to examine it, on 24 February 1616. Bellarmine, who was not one of these 11, was ordered by the Pope to convey this decision to Galileo, which he did verbally on 26 February 1616. The Decree of the Index was issued on 5 March 1616 in which "…the books by Nicolaus Copernicus and Diego Zúñiga be suspended until corrected…" This decree was signed by the Most Illustrious and Reverend Lord Cardinal of St. Cecilia, Bishop of Albano P. (Paolo Sfondrati) and Fra Francisco Magdelenus Capiferreus, O.P., Secretary. *Thony Christie, My thanks to Thony for the correction More detail about this event can be found on the Feb 26 Post about Galileo

Original 1543 Nuremberg edition of De revolutionibus orbium coelestium (English translation: On the Revolutions of the Heavenly Spheres)




1639 Debeaune to Mersenne: “I do not think that one could acquire any solid knowledge of nature in physics without geometry, and the best of geometry consists of analysis, of such kind that without the latter it is quite imperfect.” *VFR


1673 Hooke presents Arithmetic Engine to Royal Society. After a presentation of a calculating machine by Leibniz on January 22, (after which Leibniz complained to Oldgenburg that Hooke's examination of the machine had shown "almost indecent interest") Hooke became interested in creating a better machine and announced such intention to the Royal Society. Working with Richard Shortgrave, Harry Hunt and John Pell he produced a machine which would multiply to twenty places over the next six weeks. His diary entry seemed to indicate the demonstration went well, but within a few days he seemed to have dismissed such machines entirely. *Stephen Inwood, Forgotten Genius
Image of Leibniz calculator: 




1684 Halley's father mysteriously went missing and five weeks later was found murdered on the banks of the Medway. *Kate Morant, halleyslog.wordpress.com


On March 5, 1750, Euler read his own Recherches sur la Précession at the Berlin Academy. Two days later he wrote d'Alembert giving an extended account of his struggle to derive the precession and giving d'Alembert credit for re-inspiring his efforts to solve it. * Curtis Wilson, Historia Mathematica, Volume 35, Issue 4, November 2008, Pages 329–332




1831 Birth of "The Average Man". Adolphe Quetelet read a memoir to the Brussels Academy Royal. The newborn l'homme moyen would not be officially named by Quetelet until July. *Statistics on the Table: The History of Statistical Concepts and Methods By Stephen M. Stigler  

image: First edition of Quetelet's principal work in which he presented his conception of the homme moyen (“average man”) as the central value about which measurements of a human trait are grouped according to the normal distribution. Sur l’Homme et le Développement de ses Facultés, ou Essai de Physique Sociale. Lambert Adolphe Jacques Quetelet.



On this day in 1835 a ceremony to honor The Genius and Discoveries of Sir Isaac Newton was organized by the citizens of the Lincolnshire, his area of birth, a few years after the centennial of his death. By unanimous choice, the committee selected as the speaker, the 19 year-old George Boole.

All present were struck by the youthful age of the speaker and not a little amazed by both his knowledge of the subject and his confident lecturing style.  *SAU



1876 Sylvester, at age 61, appointed professor of mathematics at Johns Hopkins University. This was the real beginning of graduate mathematics education in the United States. *VFR


1960 Gao–Guenie (H5 ordinary chondrite) meteorites fell in Burkina Faso on March 5, 1960 at 17:00 (local time). After three separate detonations, several thousands of stones rained down over an area of about 70 square kilometres (27 sq mi). The sound of the fall was heard as far as Ouagadougou, which is 100 kilometers (62 mi) away. Eyewitnesses said that some trees were broken and henhouses destroyed. The largest stones recovered weigh up to 10 kilograms (22 lb)*Wik 



1963 On this day in 1963, the Hula-Hoop, a hip-swiveling toy that became a huge fad across America when it was first marketed by Wham-O in 1958, is patented by the company’s co-founder, Arthur “Spud” Melin. An estimated 25 million Hula-Hoops were sold in its first four months of production alone. *http://www.history.com



1981  Today in 1981  the ZX81, a pioneering British home computer, is launched by Sinclair Research and would go on to sell over 1 1⁄2 million units around the world.  The ZX81 is a home computer that was produced by Sinclair Research and manufactured in Dundee, Scotland, by Timex Corporation. It was launched in the United Kingdom in March 1981 as the successor to Sinclair's ZX80 and designed to be a low-cost introduction to home computing for the general public. It was hugely successful; more than 1.5 million units were sold. In the United States it was initially sold as the ZX-81 under licence by Timex. It had a smashing 1Kb of Ram.  



1993 Talking Laptop Helps Blind Student Earn B.S.:
In an early demonstration of the impact computers could have on people's lives, the Los Angeles Times reports that a blind student was taking advantage of a talking laptop computer to help him complete courses necessary to graduate from UCLA. After 15 years of going to college on and off, the computer provided Robert Antunez the independence and aid he needed to complete a bachelor's degree in political science. *CHM






1995 The Yahoo! search engine officially launches on the Internet. 13 months later, Yahoo! will hold its IPO at a price of $13 per share. Yahoo!’s stock will peak at $475 in January 2000, and fall to $8.02 in September 2001.






BIRTHS

1512 Gerardus Mercator (5 Mar 1512- 2 Dec 1594) Flemish cartographer whose most important innovation was a map, embodying what was later known as the Mercator projection, on which parallels and meridians are rendered as straight lines spaced so as to produce at any point an accurate ratio of latitude to longitude. He also introduced the term atlas for a collection of maps. *TIS A nice blog about the Mercator projection, which he suggests should be called the Mercator Wright projection is at the Renaissance Mathematicus blogsite.
For those interested in a quick look at the math involved in the Mercator-Wright projection, this Endeavour blog by John D. Cook may help.

Mercator 1569 world map (Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata) showing latitudes 66°S to 80°N





1575 William Oughtred (5 Mar 1575; 30 Jun 1660 at age 85) English mathematician and Episcopal minister who invented the earliest form of the slide rule, two identical linear or circular logarithmic scales held together and adjusted by hand. Improvements involving the familiar inner rule with tongue-in-groove linear construction came later. He also introduced the familiar multiplication sign x in a 1631 textbook, along with the first use of the abbreviations sin, cos and tan.*Tis There is an Oughtred Society dedicated to the history and preservation of slide rules.

William Oughtred's most important work was first published in 1631, in Latin, under the title Arithemeticæ in Numeris et Speciebus Institutio, quae tum Logisticæ, tum Analyticæ, atque adeus totius Mathematicæ quasi Clavis est (i.e. "The Foundation of Arithmetic in Numbers and Kinds, which is as it were the Key of the Logistic, then of the Analytic, and so of the whole Mathematic(s)"). It was dedicated to William Howard, son of Oughtred's patron Thomas Howard, 14th Earl of Arundel.

This is a textbook on elementary algebra. It begins with a discussion of the Hindu-Arabic notation of decimal fractions and later introduces multiplication and division sign abbreviations of decimal fractions. Oughtred also discussed two ways to perform long division and introduced the "~" symbol, in terms of mathematics, expressing the difference between two variables. Clavis Mathematicae became a classic, reprinted in several editions. It was used as a textbook by John Wallis and Isaac Newton among others. A concise work, it argued for a less verbose style in mathematics, and greater dependence on symbols. 

The first edition of John Wallis's foundational text on infinitesimal calculus, Arithmetica Infinitorum (1656), carries a long letter of dedication to William Oughtred.






1624/25 John Collins (5 March 1624 in Wood Eaton (4km north of Oxford), England - 10 Nov 1683 in London, England) was an accountant and publisher who corresponded extensively with the mathematicians of his day. Collins's importance is, as Barrow said, being "the English Mersenne" . He corresponded with Barrow, David Gregory, James Gregory, Newton, Wallis, Borelli, Huygens, Leibniz, Tschirnhaus and Sluze.
Collins published books by Barrow and Wallis and left a collection of 2000 books and an uncounted number of manuscripts.
He did publish works of his own, however. For instance he published works on sundials, trigonometry for navigation and the use of the quadrant. He had a paper on cartography published and also wrote on accounting, compound interest and annuities. His major works were An introduction to merchant's accounts (1652), The sector on a quadrant (1658), Geometrical dialling (1659), The mariner's plain scale new plained (1659) and, in 1664, he published Doctrine of Decimal Arithmetick. *SAU

About twenty-five years after Collins's death his books and papers came into the possession of William Jones, F.R.S. They included a voluminous correspondence with Newton, Leibniz, Gregory, Barrow, John Flamsteed, Wallis, Slusius, and others. From it was selected and published in 1712, by order of the Royal Society, the Commercium Epistolicum, of material relevant to Newton's priority over Leibniz in the discovery of the infinitesimal calculus; specimens of results from the use of the fluxional method were transmitted 20 July 1669 through Barrow to Collins, and by him made widely known. *Wik





1779 Benjamin Gompertz (March 5, 1779 – July 14, 1865), was a self educated mathematician, denied admission to university because he was Jewish.  Nevertheless he was made Fellow of the Royal Society in 1819. Gompertz is today mostly known for his Gompertz law (of mortality), a demographic model published in 1825. The model can be written in this way:

N(t) = N(0) e-c (e{at}-1),

where N(t) represents the number of individuals at time t, and c and a are constants.

This model is a refinement of the demographic model of Malthus. It was used by insurance companies to calculate the cost of life insurance. The equation, known as a Gompertz curve, is now used in many areas to model a time series where growth is slowest at the start and end of a period. The model has been extended to the Gompertz–Makeham law of mortality.


1794 Jacques Babinet (5 March 1794 – 21 October 1872) was a French physicist, mathematician, and astronomer who is best known for his contributions to optics. A graduate of the École Polytechnique, which he left in 1812 for the Military School at Metz, he was later a professor at the Sorbonne and at the Collège de France. In 1840, he was elected as a member of the Académie Royale des Sciences. He was also an astronomer of the Bureau des Longitudes.
Among Babinet's accomplishments are the 1827 standardization of the Ångström unit for measuring light using the red Cadmium line's wavelength, and the principle (Babinet's principle) that similar diffraction patterns are produced by two complementary screens. He was the first to suggest using wavelengths of light to standardize measurements. His idea was first used between 1960 and 1983, when a meter was defined as a wavelength of light from krypton gas.
In addition to his brilliant lectures on meteorology and optics research, Babinet was also a great promoter of science, an amusing and clever lecturer, and a brilliant, entertaining and prolific author of popular scientific articles. Unlike the majority of his contemporaries, Babinet was beloved by many for his kindly and charitable nature. He is known for the invention of polariscope and an optical goniometer. *Wik

The polariscope is an optical inspection device used to detect internal stresses in glass and other transparent materials such as plastics.   A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (gōnía) 'angle' and μέτρον (métron) 'measure'. The protractor is a commonly used type in the fields of mechanics, engineering, and geometry.

The first known description of a goniometer, based on the astrolabe, was by Gemma Frisius in 1538.






1815 Angelo Genocchi (5 March 1817 – 7 March 1889) was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi numbers are named after him. G(t)= 2t/(et+1)for integer values of t. The first few are 1, −1, 0, 1, 0, −3, 0, 17...(A001469 in OEIS)
Genocchi was President of the Academy of Sciences of Turin.*Wik  The unsigned coefficients of Genocchi numbers give expansion of x*tan(x/2). *PB




1887  Otto Haupt (born 5 March 1887 in Würzburg; died 10 November 1988 in Bad Soden) was a German mathematician.
Haupt obtained his PhD in 1911 under the supervision of Georg Rost and Emil Hilb at the University of Würzburg, and became a professor at the University of Erlangen-Nuremberg. He retired from teaching in 1953, but continued his mathematical research for many subsequent years.
In 1918 he was married to Edith Hughes. Despite her Jewish ancestry, she survived the Nazi period unharmed in Erlangen, and lived to 1981.

Haupt specialized in geometry and real analysis; many of his research publications related to the four-vertex theorem on local minima and maxima of curvature. He also wrote textbooks on algebra and calculus.
In 1987, his centenary year, a birthday conference was given in his honor at the University of Erlangen.
He was awarded honorary doctorates from the University of Bonn, the University of Nantes and the University of Würzburg.





1915 Laurent-Moïse Schwartz (5 March 1915 in Paris – 4 July 2002 in Paris) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work (developing the theory of distributions, a new notion of generalized functions motivated by the Dirac delta-function of theoretical physics). (Harald Bohr presented the Fields Medal to Schwartz at the International Congress of Mathematicians in Harvard).
He was the first French mathematician to receive the Fields medal. For a long time he taught at the École polytechnique. *Wik



1943 Elizabeth Ruth Naomi Belville (5 March 1854 – 7 December 1943), also known as the Greenwich Time Lady, was a businesswoman from London. She, her mother Maria Elizabeth, and her father John Henry, sold people the time. This was done by setting a watch to Greenwich Mean Time, as shown by the Greenwich clock, and then selling people the time by letting them look at the watch. *Wik A nice blog about time, and the time lady by Greg Ross at Futility Closet. and a book by David Rooney.

Ruth Belville, the "Time Lady," died Dec. 7, 1943, at the age of 89. For almost 50 years, Ruth sold the time to a select clientele in London. She would travel to Greenwich Observatory every Tuesday, where she would synchronize the family chronometer, a large pocket watch, to the master clock at Greenwich. Then she would take the train to London, dropping in one by one on her subscribers, who for various reasons needed to know the exact time, so they could set their timepieces from hers. She inherited this odd profession, and the chronometer, from her father, John Henry, who began distributing time in 1836, and then her mother Maria, who continued the practice after her father's death, until she retired in 1892 and Ruth took over.

When the time-sharing business began in 1836, the only way to know the exact time, unless you were inside the Observatory looking at the master clock, was to wait for the time ball to drop to the Observatory roof at exactly 1:00 PM each day. That was fine if you were on a ship on the Thames, but hardly of use to a watchmaker in London. John Henry, who was in charge of the time ball, was instructed by the Astronomer Royal to carry the exact time to London once or twice a week, so that London clockmakers and railroad managers could ensure that their timepieces were accurate. It is usually said that John Henry carried the time himself, but that is hardly likely, as he was one of the busiest employees at the Observatory, so he probably just set the watch and had a carrier take it to London.

Linda Hall Org


1880 Sergei Natanovich Bernstein (March 5, 1880 – October 26, 1968) was a Russian and Soviet mathematician. His doctoral dissertation, submitted in 1904 to the Sorbonne, solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. Later, he published numerous works on Probability theory, Constructive function theory, and mathematical foundations of genetics. From 1906 until 1933, Bernstein was a member of the Kharkov Mathematical Society. *Wik

He interrupted his studies in France to spend three terms at the University of Göttingen, beginning in the autumn of 1902, where his studies were supervised by David Hilbert.

Hilbert stated his nineteenth problem as a regularity problem for a class of elliptic partial differential equation with analytic coefficients. Therefore the first efforts of researchers who sought to solve it were aimed at studying the regularity of classical solutions for equations belonging to this class. For C 3  solutions, Hilbert's problem was answered positively by Sergei Bernstein (1904) in his thesis.  

.On the other hand, direct methods in the calculus of variations showed the existence of solutions with very weak differentiability properties. For many years there was a gap between these results. The solutions that could be constructed were known to have square integrable second derivatives, but this was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958), who were able to show the solutions had first derivatives that were Hölder continuous. By previous results this implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem. Subsequently, Jürgen Moser gave an alternate proof of the results obtained by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958)




1885 Pauline Sperry (March 5, 1885 – September 24, 1967)  born in Peabody, Massachusetts. After graduating Phi Beta Kappa from Smith College in 1906 she taught several years before doing graduate work at the University of Chicago under the projective differential geometer Ernest Julius Wilczynski (1876–1932). Her doctoral thesis, "Properties of a certain projectively defined two-parameter family of curves on a general surface", drew on his work as the founder of the American school of projective differential geometry. After receiving her Ph.D. in 1916 she taught at the University of California at Berkeley, becoming the first woman to be promoted to assistant professor in mathematics (in 1923). In 1950 she was fired for refusing to sign a loyalty oath.  

At the height of McCarthyism, the Board of Regents required university employees to sign a loyalty oath. Sperry, Hans Lewy, and others who refused were barred from teaching without pay in 1950. In the case Tolman v. Underhill, the California Supreme Court ruled in 1952 the loyalty oath unconstitutional and reinstated those who refused to sign. Sperry was reinstated with the title emeritus associate professor and later awarded back pay. *Wik




1915 Laurent-Moïse Schwartz (5 March 1915 in Paris – 4 July 2002 in Paris) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work (developing the theory of distributions, a new notion of generalized functions motivated by the Dirac delta-function of theoretical physics). He was the first French mathematician to receive the Fields medal. For a long time he taught at the École polytechnique. *Wik




1931 Vera S. Pless (nee Stepen; March 5, 1931 – March 2, 2020)  is an American mathematician specializing in combinatorics and coding theory. She was professor emeritus at the University of Illinois at Chicago. She has co-authored several articles with John H. Conway, giving her an Erdős number of 2.

As a teenager, she was more interested in playing the cello than in mathematics, but she left high school two years early to go to the University of Chicago, and finished her studies there in three years.

Inspired by Irving Kaplansky to study abstract algebra, she stayed at the university for a master's degree, which she earned in 1952 not long after marrying her husband, a high-energy experimental physicist.

Two years later, bored with being a stay-at-home mother, Pless began teaching courses at Boston University, and a few years later began searching for a full-time job. Unable to obtain an academic position, she took a position at the Air Force Cambridge Research Laboratory in Massachusetts. where she began working on error-correcting codes.

She returned to Chicago in 1975 as a full professor of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago. Her husband and youngest son had remained in the Boston area, and five years after the move, she and her husband divorced.

She retired in 2006 and died at her home in Oak Park, Illinois on March 2, 2020 at the age of 88.*Wik


*AMS




DEATHS

1827 Pierre Simon, Marquis de Laplace (23 Mar 1749, 5 Mar 1827 at age 78) was a French mathematician, physicist, statistician and astronomer known for his mathematical analysis of the stability of the solar system (1773), alleviating Isaac Newton's concerns about perturbations between planets. He took an exact approach to science. He developed an explanation of surface tension of a liquid in terms of inter-molecular attractions, investigated capillary action and the speed of sound. He assisted Antoine Lavoisier (1783) investigating specific heat and heats of combustion, initiating the science of thermochemistry. He believed the solar system formed from a collapsing nebula. He contributed to the mathematics of probability and calculus, in which a differential equation is known by his name, and was involved in establishing the metric system.*TIS His last words were, “What we know is very slight; what we don’t know is immense.” *Eves, Revisited, 319◦

The first American translation of his classic Traité de mécanique céleste was done by Nathanial Bowditch. The work was twelve volumes long by the time it was completed by Laplace, the first four volumes  extended to 1508 quarto (small) pages.  By the time Bowditch completed his translation of the four volumes, explaining the work took 3832 large pages.  Perhaps we can now more clearly understand Bowditch's famous quote, "Whenever I meet in La Place with the words 'Thus it plainly appears,' I am sure that hours, and perhaps days, of hard study will alone enable me to discover how it plainly appears."

"

*Wik





1827 Count Alessandro Giuseppe Antonio Anastasio Volta (18 Feb 1745; 5 Mar 1827 at age 82) Italian physicist who invented the electric battery (1800), which for the first time enabled the reliable, sustained supply of current. His voltaic pile used plates of two dissimilar metals and an electrolyte, a number of alternated zinc and silver disks, each separated with porous brine-soaked cardboard. Previously, only discharge of static electricity had been available, so his device opened a new door to new uses of electricity. Shortly thereafter, William Nicholson decomposed water by electrolysis. That same process later enabled Humphry Davy to isolate potassium and other metals. Volta also invented the electrophorus, the condenser and the electroscope. He made important contributions to meteorology. His study of gases included the discovery of methane. The volt, a unit of electrical measurement, is named after him.*TIS




1875 Claude-Louis Mathieu (25 Nov 1783; 5 Mar 1875) French astronomer and mathematician who worked particularly on the determination of the distances of the stars. He began his career as an engineer, but soon became a mathematician at the Bureau des Longitudes in 1817 and later professor of astronomy in Paris. For many years Claude Mathieu edited the work on population statistics L'Annuaire du Bureau des Longitudes produced by the Bureau des Longitudes. His work in astronomy focussed on determining the distances to stars. He published L'Histoire de l'astronomie au XVIII siècle in 1827. *TIS



1982  Karol Borsuk (May 8, 1905, Warsaw – January 24, 1982, Warsaw) Polish mathematician. His main interest was topology.
Borsuk introduced the theory of absolute retracts (ARs) and absolute neighborhood retracts (ANRs), and the cohomotopy groups, later called Borsuk-Spanier cohomotopy groups. He also founded the so called Shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century. *Wikipedia


1925 Johan Ludwig William Valdemar Jensen (8 May 1859 in Nakskov, Denmark - 5 March 1925 in Copenhagen, Denmark)contributed to the Riemann Hypothesis, proving a theorem which he sent to Mittag-Leffler who published it in 1899. The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped. It expresses, "... the mean value of the logarithm of the absolute value of a holomorphic function on a circle by means of the distances of the zeros from the center and the value at the center. "
He also studied infinite series, the gamma function and inequalities for convex functions.*SAU

*Wik



1840 Franz Carl Joseph Mertens (20 March 1840 in Schroda, Posen, Prussia (now Środa Wielkopolska, Poland) - 5 March 1927 in Vienna, Austria) Mertens worked on a number of different topics including potential theory, geometrical applications to determinants, algebra and analytic number theory, publishing 126 papers. Bruce C Berndt writes, "Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name. "
Many people are aware of Mertens contributions since his elementary proof of the Dirichlet theorem appears in most modern textbooks. However he made many deep contributions including Mertens' theorems, three results in number theory related to the density of the primes. He proved these results using Chebyshev's theorem, a weak version of the prime number theorem. *SAU
In his youth, Mertens moved to Berlin where he became a student at Berlin
University, and where he studied under Kronecker and Kummer. Mertens first worked in Krakow, and then moved to Austria. Ernst Fischer and Schrodinger, for instance, were students of Mertens at the University of Vienna. *Julio Gonzalez Cabillon, Historia Matematica Discussions



1885 John Radford Young (1799– March 5,1885; Peckam, England) was a mathematician, professor and author, who was almost entirely self-educated. At an early age he became acquainted with Olinthus Gilbert Gregory, who perceived his mathematical ability and assisted him in his studies.
In 1833, he was appointed Professor of Mathematics at Belfast College. When Queen's College, Belfast, opened in 1849, the Presbyterian party in control there prevented Young's reappointment as Professor in the new establishment. From that time he devoted himself more completely to the study of mathematical analysis, and made several original discoveries. He appears to have been the first to use the term "circular function" when he used it in 1831 in the an edition of Elements of the Differential Calculus "Thus, ax, a log x, sin x, &c., are transcendental functions: the first is an exponential function, the second a logarithmic function, and the third a circular function"
In 1847, he published in the Transactions of the Cambridge Philosophical Society a paper "On the Principle of Continuity in reference to certain Results of Analysis", and, in 1848, in the Transactions of the Royal Irish Academy a paper "On an Extension of a Theorem of Euler". As early as 1844, he had discovered and published a proof of Newton's rule for determining the number of imaginary roots in an equation. In 1866, he completed his proof, publishing in The Philosophical Magazine a demonstration of a principle which in his earlier paper he had assumed as axiomatic. In 1868, he contributed to the Proceedings of the Royal Irish Academy a memoir "On the Imaginary Roots of Numerical Equations".
*Wik



1930 Christine Ladd-Franklin (1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst. She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S. Peirce’s logic and she was highly praised by Prior.

In 1878, Ladd was accepted into Johns Hopkins University with the help of James J. Sylvester, an English mathematician among the university's faculty who remembered some of Ladd's earlier works in the Educational Times. Ladd's application for a fellowship was signed "C. Ladd", and the university offered her the position without realizing she was a woman.[8] When they did realize her gender, the board tried to revoke the offer, but Sylvester insisted that Ladd should be his student, and so she was.[8] She held a fellowship at Johns Hopkins University for three years, but the trustees did not allow her name to be printed in circulars with those of other fellows, for fear of setting a precedent.[8] Furthermore, dissension over her continued presence forced one of the original trustees to resign. *Wik

Sylvester's letter in support of Ladd





1954 Julian Lowell Coolidge (28 Sep 1873, 5 Mar 1954 at age 80) American mathematician and educator who published numerous works on theoretical mathematics along the lines of the Study-Segre school. Coolidge received a B.A. at Harvard (1895), then in England he graduated (1897) with a B.Sc. from Balliol College Oxford. (It is interesting that this degree from Oxford was in natural science and it was the first natural science degree ever awarded by Oxford.) He taught at Groton School, Conn. (1897-9) where one of his pupils was Franklin D Roosevelt, the future U.S. president. From 1899 he taught at Harvard University. Between 1902 and 1904, he went to Turin to study under Corrado Segre and then to Bonn where he studied under Eduard Study. His Mathematics of the Great Amateurs is perhaps his best-known work. *TIS . This geometer wrote several noteworthy books on the history of geometry.*VFR





1990 Edgar Raymond Lorch (July 22, 1907 – March 5, 1990) was a Swiss American mathematician. Described by The New York Times as "a leader in the development of modern mathematics theory", he was a professor of mathematics at Columbia University. He contributed to the fields of general topology, especially metrizable and Baire spaces, group theory of permutation groups and functional analysis, especially spectral theory, convexity in Hilbert spaces and normed rings.

Born in Switzerland, Lorch emigrated with his family to the United States in 1917 and became a citizen in 1932. He joined the faculty of Columbia University in 1935 and retired in 1976, although he continued to write and lecture as professor emeritus. For his reminiscences of Szeged, Lorch posthumously received in 1994 the Lester R. Ford Award, with Reuben Hersh as editor.*Wik 




2018 Clarence Francis Stephens (July 24, 1917 – March 5, 2018) was an American mathematician. He is credited with inspiring students and faculty at SUNY Potsdam to form the most successful United States undergraduate mathematics degree programs in the past century.[citation needed] Stephens was recognized by Mathematically Gifted & Black as a Black History Month 2018 Honoree.

The fifth of six children, he was orphaned at the age of eight. For his early education, he attended Harbison Agricultural and Industrial Institute, a boarding school for African-Americans in Irmo, South Carolina under Dean R. W. Bouleware and later President Rev John G. Porter.

Stephens graduated from Johnson C. Smith University in 1938 with a B.S. degree in mathematics. He received his M.S. (1939) and his Ph.D. (1944) from the University of Michigan, with a thesis on Non-Linear Difference Equations Analytic in a Parameter under James Nyswander.

After serving in the U.S. Navy (1942–1946) as a Teaching Specialist, Dr. Stephens joined the mathematics faculty of Prairie View A&M University. The next year (1947) he was invited to join the mathematics faculty at Morgan State University.

As a Mathematics Association of America (MAA) biography explains, “Dr. Stephens' focus was on being a research mathematician, so he accepted the position in part because he would be near a research library at Johns Hopkins University. While at Morgan State University, Dr. Stephens became appalled at what a poor job was being done in general to teach and inspire students to learn mathematics. He changed his focus from being a researcher to achieving excellence, with desirable results, in teaching mathematics.

In 1953, he received a one-year Ford Fellowship to study at the Institute for Advanced Study in Princeton, New Jersey.

Dr. Stephens remained at Morgan State until 1962, where is credited with initiating the program which led to five students achieving 91% to 99% on the graduate record exam in mathematics, three of these students (Earl R. Barnes, Arthur D. Grainger and Scott W. Williams) became the only three students of the same class at a Historically Black College to earn a PhD in mathematics. Stephens accepted an appointment as professor of mathematics at SUNY Geneseo. In 1969 he left Geneseo to join the mathematics faculty at SUNY Potsdam, where he served as chair of the mathematics department until his retirement in 1987.

The MAA biography reports that during Dr. Stephens’ tenure at SUNY Potsdam "the department became nationally known as a model of teaching excellence in mathematics. For several of these years the program was among the top producers of mathematics majors in the country. The teaching techniques that Professor Stephens introduced at Potsdam, and earlier at Morgan State, have been adopted by many mathematics departments across the country. They have been described in publications by the MAA, and recently in a book, Math Education At Its Best: The Potsdam Model, by Datta (Center for Teaching/Learning of Mathematics, 1993)." He turned 100 in July 2017[4] and died in March 2018. *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