## Monday 8 May 2023

### On This Day in Math - May 8

A small circle is quite as infinite as a large circle.
G. K. Chesterton

The 128th day of the year; 128 is The largest known even number that can be expressed as the sum of two primes in exactly three ways. (Find them) *Prime Curios   How many smaller numbers (and which) are there that can be so expressed?

But, it can not be expressed as the sum of distinct squares, for any number of squares.
And it is the largest such number, ever.... no, I mean EVER.  The very last.
(Surprisingly, there are only 31 numbers that can not be expressed as the sum of distinct squares. )

128 can be expressed by a combination of its digits with mathematical operators thus 128 = 28 - 1, making it a Friedman number in base 10 (Friedman numbers are named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.)

128 the sum of the factorials of the first three prime numbers, 2! + 3! + 5! =128.

128 = 2^8, so in binary it is a 1 followed by 7 zeros, which makes it also 4^4, and in base 4 its a 2 with three zeros.  But it's also 8^2, so in base eight its a 2 with two zeros,
128 is a power of two, and all of its digits are powers of two.  I don't know of any other.

128 can be expressed by a combination of its digits with mathematical operators thus 128 = 28 - 1, making it a Friedman number in base 10 (Friedman numbers are named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.)

128 the sum of the factorials of the first three prime numbers, 2! + 3! + 5! =128.

Some nice relationships between 128 and its digits, 128 + (1+2+8) = 139, a prime number.  But 128 + (8 + 1) is 137, also prime, and 128 + (2 + 1)  is 131, a prime, AND 128 +( 8+2 ) is not prime, but 138 is between a twin prime pair.  ..... And 1*2*8 = 16 is a divisor of 128.
And that pair of cousin primes, 127 and 131, are the largest  such pair with a power of two (128) between them.

The name for a particular 7th dimensional Hyperplex with 128 vertices is a Hepteract.  Dazzle your friends.

Oh, I told you 128 is the 7th power of two.... but there are no more three digit numbers that are 7th powers...

And if you like to keep score, 128 is 6 score and 8.  In old commercial terminology, a schock was a lot of 60 items, so 128 is also two shock and 8, or 28 in sexigesimal (base sixty).   The number of  stalks of corn or wheat (supposedly)  gathered and stood on ends in the fields to dry, like in "When the frost is on the Pumpkin and the Fodders in the shock. "

128  is divisible by four so it is the difference of two squares of numbers that differ by 2, and since  128 / 4 = 32, the numbers must straddle 31, 33² - 31² = 128.
But it is also divisible by eight, so it is the difference of two squares of numbers that differ by four(there is a power of two relation working here, which students might find).  And since 128/8 = 16, 18² - 14² = 128

And in 1968 the 128 K Mac was the hottest desktop computer around.

EVENTS

1654 Otto von Guericke demonstrates the Magdenburg hemispheres in front of the imperial Diet, and the Emperor Ferdinand IIII in Regensburg.
The Magdeburg hemispheres, around 50 cm (20 inches) in diameter, were designed to demonstrate the vacuum pump that Guericke had invented. One of them had a tube connection to attach the pump, with a valve to close it off. When the air was sucked out from inside the hemispheres, and the valve was closed, the hose from the pump could be detached, and they were held firmly together by the air pressure of the surrounding atmosphere.
Thirty horses, in two teams of fifteen, could not separate the hemispheres until the valve was opened to equalize the air pressure. In 1656 he repeated the demonstration with sixteen horses (two teams of eight) in his hometown of Magdeburg, where he was mayor. He also took the two spheres, hung the two hemispheres with a support, and removed the air from within. He then strapped weights to the spheres, but the spheres would not budge.Gaspar Schott was the first to describe the experiment in print in his Mechanica Hydraulico-Pneumatica (1657).
In 1663 (or, according to some sources, in 1661) the same demonstration was given in Berlin before Frederick William, Elector of Brandenburg with twenty-four horses. It is unclear how strong a vacuum Guericke's pump was able to achieve, but if it was able to evacuate all of the air from the inside, the hemispheres would have been held together with a force of around 20 000 N (4400 lbf, or 2.2 short tons), equivalent to lifting a car or small elephant; a dramatic demonstration of the pressure of the atmosphere. *Wik

1661 “On 8 May 1661 the Society’s Journal Book notes that ‘a motion was made for the erecting of a library’, and later in the same month ‘it was resolved that every member, who hath published or shall publish any work, give the Society one copy’.” (from Emma Davidson at RSI)

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1698  Henry Baker was born 8 . His book The Microscope Made Easy (1743) has been described as the first laboratory manual for microscopy. *RMAT

1774 The conjunction of the Planets Jupiter, Mars, Venus, Mercury and the Moon on this date would herald the apocalypse according to a treatise by Eelco Alta, a Frisian clergyman and theologian. However, the apocalypse did not occur, perhaps because the projected conjunction of the heavenly bodies never occurred. One good result attributed to the treatise was the creation of what is now the oldest continuously operating planetarium in the world, the Eise Eisinga Planetarium in the ceiling of his former home in the Netherlands. It is driven by a pendulum clock, which has 9 weights or ponds. The planets move around the model in real time, automatically. (A slight "re-setting" must be done by hand every four years to compensate for the February 29th of a leap year.) In addition to the basic orrery, there are displays of the phase of the moon and other astronomical phenomena. The planetarium includes a display for the current time and date. The plank that has the year numbers written on it has to be replaced every 22 years. To create the gears for the model, 10,000 handmade nails were used. *Wik *collected notes

1790 The Assembly (French) ordered the Académie des Sciences to standardize weights and measures on 8 May 1790. The Académie appointed a Commission of Lagrange, Borda, Condorcet, Laplace and Tillet to compare the decimal and duodecimal systems. Another Commission, with Monge instead of Tillet, was to examine how to make a standard of length. The Commissions continued functioning through the Revolution.

1794 Lavoisier Guillotined along with twenty-seven other members of the Ferme Générale, including his father-in-law. See Deaths below

In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

1795 French astronomer Jerome Lalande observes a "star". It is in fact the planet Neptune, which is not officially discovered until 1846. * Liz Suckow@LizMSuckow
A second recording, noting a possible error on the 10th was entered. Discovery of these recordings 1n 1947 by Sears C. Walker of the U.S. Naval Observatory led to a better calculation of the planet's orbit. *Wik

In 1886, Coca-Cola, the soft drink, was first sold to the public at the soda fountain in Jacob's Pharmacy in Atlanta, Georgia. It was invented by pharmacist, John Stith Pemberton, who mixed it in a 30-gal. brass kettle hung over a backyard fire. Until 1905, the drink, marketed as a "brain and nerve tonic," contained extracts of cocaine as well as the caffeine-rich kola nut. The name, using two C's from its ingredients, was suggested by his bookkeeper Frank Robinson, whose excellent penmanship provided the first scripted "Coca-Cola" letters as the famous logo. Asa Candler marketed Coke to world after buying the company from Pemberton. *TIS

 John Pemberton, *Wik

1910 The New York Times Sunday Magazine publishes banner headline, "Fears Of The Comet Are Foolish And Ungrounded," only ten days before the Earth moves into the tail of Halley's Comet. The article featured the famous female astronomer, Mary Proctor, debunking horror stories such as :

Here is a gigantic monster in the sky with a head over two hundred thousand miles in width… and a train two million miles in length, rushing through space at the alarming rate of a thousand miles a minute.
On May 18 the earth will be plunged in this white hot mass of glowing gas, and, according to the report of the ignorant and superstitious, the world will be set on fire.
These sensation makers further say that the oceans on the side facing the comet will be boiled by the intense heat, and the land scorched and blistered as the dread wanderer passes by on its baneful way.

*SundayMagazine.org

Halley’s comet approached Earth and killed England’s King Edward VII, according to some superstitious folk. No one could definitively say how it did, but it certainly did. And that wasn’t its only offense. The Brits also figured it was an omen of a coming invasion by the Germans, while the French reckoned it was responsible for flooding the Seine.

A French Scientist named Camille Flammarion, in typical French despair, reckoned that as we passed through the comet’s tail, “cyanogen gas would impregnate the atmosphere and possibly snuff out all life on the planet,” *Wired

1932 The USS Akron, an American dirigible and the world's first purpose-built flying aircraft carrier, flew mail from Lakehurst, New Jersey,  to San Diego, On This Day in 1932. The ship reached Camp Kearny in San Diego, on the morning of 11 May and attempted to moor. Since neither trained ground handlers nor specialized mooring equipment were present, the landing at Camp Kearny was fraught with danger. By the time the crew started the evaluation, the helium gas had been warmed by sunlight, increasing lift. Lightened by 40 short tons (36 t), the amount of fuel spent during the transcontinental trip, the Akron was now all but uncontrollable.

The mooring cable was cut to avert a catastrophic nose-stand by the errant airship which floated upward. Most of the mooring crew—predominantly "boot" seamen from the Naval Training Station San Diego—released their lines although four did not. One let go at about 15 ft (4.6 m) and suffered a broken arm while the three others were carried further aloft. Of these Aviation Carpenter's Mate 3rd Class Robert H. Edsall and Apprentice Seaman Nigel M. Henton soon plunged to their deaths while Apprentice Seaman C. M. "Bud" Cowart held on to his line until being hoisted on board the airship an hour later. The Akron moored at Camp Kearny later that day before proceeding to Sunnyvale, California. The deadly accident was recorded on newsreel film. *Postal Museum @PostalMuseum *Wik

1961 President J. F. Kennedy presented astronaut Alan B. Shepard the ﬁrst National Aeronautics and Space Administration Distinguished Flying Medal for making America’s ﬁrst space ﬂight on May 5, 1961.*VFR

 Shephard and Capsule recovered after splashdown

BIRTHS

1859 Johan Ludwig William Valdemar Jensen (8 May 1859 – 5 March 1925) 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 centre and the value at the centre.
He also studied infinite series, the gamma function and inequalities for convex functions.*SAU

1905 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. *Wik

DEATHS

1794 Antoine Laurent Lavoisier (26 August 1743 – 8 May 1794) after a trial that lasted less than a day, a revolutionary tribunal condemned Antoine Laurent Lavoisier to death. He was 51 and guillotined on the same afternoon.  " It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like." Joseph Louis Lagrange, the day of Lavoisier’s execution.
Lavoisier was guillotined in the terror following the French Revolution. In 1778, he found that air consists of a mixture of two gases which he called oxygen and nitrogen. By studying the role of oxygen in combustion, he replaced the phlogiston theory. Lavoisier also discovered the law of conservation of mass and devised the modern method of naming compounds, which replaced the older nonsystematic method. Under the Reign of Terror, despite his eminence and his services to science and France, he came under attack as a former Ferme Générale. In November 1793, all former members of the Ferme Générale including Lavoisier and his father-in-law, were arrested and imprisoned. After a trial that lasted less than a day, they were all found guilty of conspiracy against the people of France and condemned. When Lavoisier requested time to complete some scientific work, the presiding judge was said to have answered "The Republic has no need of scientists." He was guillotined and thrown in a common grave in the Cimetière de Picpus. Mathematician Joseph Louis Lagrange lamented the execution: "It took them only an instant to cut off that head, but France may not produce another like it in a century." About eighteen months following his death, Lavoisier was exonerated by the French government. When his belongings were delivered to his widow, a brief note was included reading "To the widow of Lavoisier, who was falsely convicted."
For more about Lavoisier see SomeBeans blog

1853 John Farrar (July 1, 1779 – May 8, 1853) died at Cambridge, Massachusetts.His translations from the French, including Legendre’s Elements of Geometry (Boston, 1819),were widely used in the U. S. *VFR
He first coined the concept of hurricanes as “a moving vortex and not the rushing forward of a great body of the atmosphere”, after the Great September Gale of 1815. Farrar
remained Professor of Mathematics and Natural Philosophy at Harvard University between 1807 and 1836. During this time, he introduced modern mathematics into the curriculum. He was also a regular contributor to the scientific journals. *Wikipedia

He is buried in MountAuburn Cemetery Cambridge, Massachusetts,USA.*Wik

1904 Eadweard Muybridge English photographer important for his pioneering work in photographic studies of motion and in motion-picture projection. For his work on human and animal motion, he invented a superfast shutter. Leland Stanford, former governor of California, hired Muybridge to settle a hotly debated issue: Is there a moment in a horse’s gait when all four hooves are off the ground at once? In 1972, Muybridge took up the challenge. In 1878, he succeeded in taking a sequence of photographs with 12 cameras that captured the moment when the animal’s hooves were tucked under its belly. Publication of these photographs made Muybridge an international celebrity. Another noteworthy event in his life was that he was tried (but acquitted) for the murder of his wife's lover. *TIS

1951 Gilbert Ames Bliss, (9 May 1876, Chicago – 8 May 1951, Harvey, Illinois), was an American mathematician, known for his work on the calculus of variations. Bliss once headed a government commission that devised rules for apportioning seats in the U.S. House of Representatives among the several states. *Wikipedia

1959 Renato Caccioppoli (20 January 1904 – 8 May 1959) His most important works, out of a total of around eighty publications, relate to functional analysis and the calculus of variations. Beginning in 1930 he dedicated himself to the study of differential equations, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential equations and partial differential equations.
In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two Banach spaces is invertible only if it is locally invertible and if the only convergent sequences are the compact ones.
Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two-dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, i.e. analytic functions whose domain belongs to the vector space Cn, proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a logarithmic residue formula for functions of two complex variables in 1949.
In 1935 Caccioppoli proved the analyticity of class C2 solutions of elliptic equations with analytic coefficients.
The year 1952 saw the publication of his masterwork on the area of a surface and measure theory, the article Measure and integration of dimensionally oriented sets (Misura e integrazione degli insiemi dimensionalmente orientati, Rendiconti dell'Accademia Nazionale dei Lincei, s. VIII, v.12). The article is mainly concerned with the theory of dimensionally oriented sets; that is, an interpretation of surfaces as oriented boundaries of sets in space. Also in this paper, the family of sets approximable by polygonal domains of finite perimeter, known today as Caccioppoli sets or sets of finite perimeter, was introduced and studied.
His last works, produced between 1952 and 1953, deal about a class of pseudoanalytic functions, introduced by him to extend certain properties of analytic functions.
In his last years, the disappointments of politics and his wife's desertion, together perhaps with the weakening of his mathematical vein, pushed him into alcoholism. His growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959 by a gunshot to the head did not surprise those who knew him. He died at his home in Palazzo Cellammare
In 1992 his tormented personality was remembered in a film directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano), in which he was portrayed by Carlo Cecchi. * Wik

1953 Benjamin Fedorovich Kagan (10 March 1869 in Shavli, Kovno (now Kaunas, Lithuania)
- 8 May 1953 in Moscow, USSR) Kagan worked on the foundations of geometry and his first work was on Lobachevsky's geometry. In 1902 he proposed axioms and definitions very different from Hilbert. Kagan studied tensor differential geometry after going to Moscow because of an interest in relativity.
Kagan wrote a history of non-euclidean geometry and also a detailed biography of Lobachevsky. He edited Lobachevsky's complete works which appeared in five volumes between 1946 and 1951. *SAU

1960 Henry Whitehead was a topologist and differential geometer who is best remembered for his work on homotopy equivalence. *SAU

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