Pat'sBlog

Tuesday 23 April 2024

On This Day in Math - April 23

"Whatever is worth saying,
can be stated in fifty words or less"
~ Stanislaw Ulam *bt (before twitter)

Thanks to @cytiaB for this one

The 113th day of the year; 113 is prime, its reversal (311) is prime, and the number you get by any reordering of its digits is still prime. Students might try to find other of these "absolute" or "permutable" primes.

Also the sum of the first 113 digits of e is prime. That was also true of yesterday's number, and tomorrow's. (I was just wondering to myself, what is the longest known string of consecutive n for which the first n digits of e are prime? And a similar question for pi? "Anyone...anyone??? Bueller???)

355 is almost exactly $113 \pi = 354.9999699.. $ No year day is closer,

This is the only solution to a² + b³ = c⁷ in positive integers *Fermat's Library

There are 13 consecutive divisible integers (non-primes) between 113 and 127. How far until the next streak as long, or longer?

********Find more of these at Math Day of the Year Facts. *********************

EVENTS

1635 The 1st public school in the United States, Boston Latin School, was founded. It is still enrolling students. *George Costanza

1827 Sir William Hamilton presented his Theory of Systems of Rays at the Royal Irish Academy in Dublin. Although he was still an undergraduate, only 21 years old, his work is one of the important works in optics, for it provided a single function that brings together mechanics, optics and mathematics. It led to establishing the wave theory of light, which gives that light is a form of energy that travels in waves. *TIS In it he begins by proving that a system of light rays filling a region of space can be focused down to a single point by a suitably curved mirror if and only if those light rays are orthogonal to some series of surfaces. Moreover, the latter property is preserved under reflection in any number of mirrors. Hamilton’s innovation was to associate with such a system of rays a characteristic function, constant on each of the surfaces to which the rays are orthogonal, which he employed in the mathematical investigation of the foci and caustics of reflected light.

1867, the Zoetrope was patented by William E. Lincoln of Providence, R.I. (No. 64,117). The device was the first animated picture machine. It provided an animation sequence of pictures lining the inside wall of a shallow cylinder, with vertical slits between the images. By spinning the cylinder and looking through the slits, a repeating loop of a moving image could be viewed.

In 1896, the first movie shown to a paying theatre audience in the U.S. was presented using Thomas Edison’s Vitascope. The movie had a series of short scenes, and were part of a program with other acts at Koster and Bial’s Music Hall, 34th St, New York City. Included in the film shorts were a ballet scene, a burlesque boxing match, waves on a sea shore, and a comic allegory The Monroe Doctrine,all of which were projected at about half life size.

1906 First American automobile meets the first American speed bump. In March of 1906, residents of Chatham Borough, New Jersey had begun construction of a speed control device, crosswalks that were five Inches high, constructed of flagstones and cobblestones. Their creation was a plan to slow down the "very fast pace" (10-15 miles per hour) of the new motor carriages that have begone to take over the roads of the center of town. On "April 22, 1906 with great fanfare and many spectators. Bystanders set up seating and vendors sold hot dogs and pop corn to serve the growing group of onlookers. The next day local newspapers reported on the wreckage and carnage from the newly discovered speed reducers." Here is the article from the New York Times on April 23:

There were several persons in the machine, and when the heavy rubber tires struck the elevation there was a palpitation of the machinery and the car shot up several feet in the air. Goggles, hats, a monkey wrench, sidecombs, hairpins and other articles flew in all directions. The crowd gave a cheer and decided the borough’s plan was effective. The ‘bumps' installed by the borough officials of the village of Chatham to check the speed of automobiles through the village had their first test yesterday, and proved a decided success.

The more conventional speed bumps we are familiar with were not invented until June of 1953. They were created by Nobel Prize winning physicist, Arthur Holly Compton, while he was Chancellor of Washington University in St. Louis, Missouri. *Quora.Com, Wik

1948 Contract signed by A. Nielsen for UNIVAC I. The UNIVAC I (UNIVersal Automatic Computer I) was the first commercial computer produced in the United States. It was designed principally by J. Presper Eckert and John Mauchly, the inventors of the ENIAC. Design work was begun by their company, Eckert-Mauchly Computer Corporation, and was completed after the company had been acquired by Remington Rand. (In the years before successor models of the UNIVAC I appeared, the machine was simply known as "the UNIVAC".) The image is not the computer, but the operators console... (no mouse for that monster)
The first UNIVAC was delivered to the United States Census Bureau on March 31, 1951, and was dedicated on June 14 that year. The fifth machine (built for the U.S. Atomic Energy Commission) was used by CBS to predict the result of the 1952 presidential election. With a sample of just 1% of the voting population it correctly predicted that Dwight Eisenhower would win. The UNIVAC I computers were built by Remington Rand's UNIVAC division (successor of the Eckert-Mauchly Computer Corporation, bought by Rand in 1950 which later became part of Sperry, now Unisys). *Wik

In 1962, the first American satellite to reach the moon surface, the Ranger IV, was launched at 3:50pm from Cape Canaveral, Florida. As intended, it impacted on the moon three days later at 7:50pm on 26 Apr, travelling at 5,963 mph. The launch vehicle was an Atlas-Agena B rocket, 102 feet high, 16 feet in diameter at the base. The distance the satellite would travel was about 229,541 miles. *TIS

1964 SEAC Computer Retired:
The National Bureau of Standards retires its SEAC (Standards Eastern Automatic Computer), which it built in Washington 15 years earlier as a laboratory for testing components and systems for setting computer standards. The SEAC was the first computer to use all-diode logic, a technology more reliable than vacuum tubes, and the first stored-program computer completed in the United States. Magnetic tape in the external storage units stores programming information, coded subroutines, numerical data, and output.*CHM

1973 The US issued a commemorative stamp honoring the 500th year of the birth of Copernicus who wrote the De Revolutionibus.

In 1994, physicists at the Department of Energy's Fermi National Accelerator Laboratory discovered the subatomic particle called the top quark.*TIS A quark is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei.

For some time, Gell-Mann was undecided on an actual spelling for the term he intended to coin, until he found the word quark in James Joyce's 1939 book Finnegans Wake:

– Three quarks for Muster Mark!

Sure he hasn't got much of a bark

And sure any he has it's all beside the mark.

The word quark is an outdated English word meaning to croak and the above-quoted lines are about a bird choir mocking king Mark of Cornwall in the legend of Tristan and Iseult.

Another explanation by some, Especially in the German-speaking parts of the world there is a widespread legend, however, that Joyce had taken it from the word Quark, a German word of Slavic origin which denotes a curd cheese, but is also a colloquial term for "trivial nonsense". *Wik

2012 An active sunspot period leads to incredible aurora in US Midwest. The aurora borealis put on a dazzling show in more than a dozen states Monday night, according to SpaceWeather.com.

A particularly spectacular display was seen in Fergus Falls in western Minnesota, and Douglas Kiesling was on hand to film a stunning time-lapse video of the event,

BIRTHS

1628 Johann Hudde was a Dutch mathematician who worked on maxima and minima and the theory of equations. He gave an ingenious method to find multiple roots of an equation. He worked on improving the algebraic methods of René Descartes, seeking to extend them to the solution of equations of a higher degree by applying an algorithm. He also developed an algorithm based on Fermat's method to deal with the maxima, minima and tangents to curves of algebraic functions. Later, he served as burgomaster of Amsterdam for 30 years. During this time time he made a mathematical study of annuities. Hudde continued with an interest in physics and astronomy, producing lenses and microscopes. He collaborated with Baruch Spinoza, of Amsterdam, on telescopes. Hudde determine that in a telescope, a plano-convex lenses were better than concavo-convex. *TIS He is buried in #58 in the high choir of the Oude kerk (old church) in Amsterdam. (Help, send pictures please?) Unfortunately, Donovan Carroll informed me that his stone is covered over by the choir loft. More about Hudde and the "lost calculus" here. And the Renaissance Mathematics has a nice article about Hudde's circle of associates that is both political and mathematical, and involves a violent murder....

1743 Samuel Williams (23 Apr 1743; 2 Jan 1817 at age 73) American natural philosopher and clergyman who organized the first expedition of its kind in the U.S. (departing on 9 Oct 1780) to observe a total solar eclipse in Penobscot Bay, Maine, although it was held by the British enemy. The eclipse was very slightly less than being total, and he is believed to be the first to observe the “ Baily's Beads” phenomenon seen along the sun's last sliver. Previously, with John Winthrop (under whom he studied) he travelled to St. John's, Newfoundland (1761) to observer the Transit of Venus. When Wintrop died, Williams succeeded him (1779) as the Hollis Professor of Mathematics and Natural Philosophy at Harvard University. He researched and taught astronomy, meteorology, and magnetism. He resigned in June 1788. He also engaged in state boundary surveys: NY and Mass. (1785-88), and Vermont and Canada (1795).*TIS

The Baily's beads, diamond ring or more rarely double diamond ring effects,[1] are features of total and annular solar eclipses. Although caused by the same phenomenon, they are distinct events during these types of solar eclipses. As the Moon covers the Sun during a solar eclipse, the rugged topography of the lunar limb allows beads of sunlight to shine through in some places while not in others. They are named for Francis Baily, who explained the effects in 1836.[2][3] The diamond ring effects are seen when only one or two beads are left, appearing as shining "diamonds" set in a bright ring around the lunar silhouette.*Wik

*Linda Hall Org

1853 Alphonse Bertillon (23 Apr 1853, 13 Feb 1914 at age 60) French criminologist who was chief of criminal identification for the Paris police from 1880. He developed an identification system known as anthropometry, or the Bertillon system, that came into wide use in France and other countries. The system records physical characteristics (eye colour, scars, deformities, etc.) and specified measurements (height, fingertip reach, head length and width, ear, foot, arm and finger length, etc) These are recorded on cards and classified according to the length of the head. After two decades this system was replaced by fingerprinting in the early 1900s because Bertillon measurements were difficult to take with uniform exactness, and could change later due to growth or surgery. *TIS

1856 Granville Tailer Woods (April 23, 1856 – January 30, 1910) was an American inventor who held more than 50 patents in the United States. He was the first African American mechanical and electrical engineer after the Civil War. Self-taught, he concentrated most of his work on trains and streetcars. One of his inventions is the Synchronous Multiplex Railway Telegraph, a variation of the induction telegraph that relied on ambient static electricity from existing telegraph lines to send messages between train stations and moving trains.

Granville T. Woods invented and patented Tunnel Construction for the electric railroad system and was referred to by some as the "Black Edison". Over the course of his lifetime, Granville Woods obtained more than 50 patents for inventions including an automatic brake and an egg incubator and for improvements to other technologies such as the safety circuit, telegraph, telephone, and phonograph.*Wik

1858 Max Planck, (April 23, 1858 – October 4, 1947) German physicist, born. He studied at Munich and Berlin, where he studied under Helmholz, Clausius and Kirchoff and subsequently joined the faculty.he became professor of theoretical physics (1889-1926). His work on the law of thermodynamics and the distribution of radiation from a black body led him to abandon classical Newtonian principles and introduce the quantum theory (1900), for which he was awarded the Nobel Prize for Physics in 1918. This assumes that energy is not infinitely subdivisible, but ultimately exists as discrete amounts he called quanta (Latin, "how much"). Further, the energy carried by a quantum depends in direct proportion to the frequency of its source radiation.*TIS

1910 Sheila Scott Macintyre (née Sheila Scott, April 23, 1910 - March 21, 1960) was a Scottish mathematician well known for her work on the Whittaker constant.(The constant isn't actually a known constant, but is known to be in a small interval. Macintyre lowered the upper bound and reduced the interval of uncertainty by about 10%). Macintyre is also well known for creating a multilingual scientific dictionary: written in English, German, and Russian; at the time of her death, she was working on Japanese.*Wik

1911 Felix Adalbert Behrend (23 April 1911 in Charlottenburg, Berlin, Germany -27 May 1962 in Richmond, Victoria, Australia) Behrend studied number theory for his doctorate at the University of Berlin with Erhard Schmidt as his advisor. He was awarded his doctorate in 1933 for his dissertation Über numeri abundantes. Even before the award of his doctorate he had published three papers on number theory, the first two being Über einen Satz von Herrn Jarnik (1932) and Über numeri abundantes (1932). Of course 1933, the year that Behrend was awarded his doctorate, was also the year that Hitler came to power in Germany.
Like many Germans who fled from the Nazi threat, he found himself in England which was at war with his native Germany. He continued his work on number theory and published "On obtaining an estimate of the frequency of the primes by means of the elementary properties of the integers" in the Journal of the London Mathematical Society in 1940. The fact that he was passionately anti-Nazi did nothing to help save him from being interned as an enemy alien in 1940 and he was put on the ship the Dunera bound for Australia. He served periods of internment at Hay, Orange and Tatura in Australia. His experiences in Camp 7 at Hay during 1940-41 are related in . One should not think that internment meant an end to mathematics, for he gave lecture courses at the Camp and prepared some of his younger fellow internees for mathematics examinations at the University of Melbourne.
After his release in 1942, Behrend was appointed as a tutor at the University of Melbourne. He continued his research in number theory and published On the frequency of the primes in the Journal of the Royal Society of New South Wales in 1942. This paper was a continuation of the one he had published in London two years earlier. In the following year he published a paper on a totally different topic. This was A polyhedral model of the projective plane which also appeared in the Journal of the Royal Society of New South Wales. Behrend is commemorated by the 'Behrend memorial lecture in mathematics', established at the University of Melbourne in 1963 with funds provided by his widow. *SAU

1914 Georgii Nikolaevich Polozii (23 April 1914 in Transbaikal, Russia - 26 Nov 1968 in Kiev, Ukraine) Polozii studied at Saratov University which had been founded in 1919. He graduated in 1937 and then was appointed to the teaching staff of the university. In 1949 Polozii was appointed to the University of Kiev and he remained at Kiev until his death in 1968.
Polozii's major pure mathematical contributions were to the theory of functions of a complex variable, approximation theory, and numerical analysis. He also made major contributions to mathematical physics and applied mathematics in particular working on the theory of elasticity.

Between 1962 and 1966 Polozii developed the theory for a new class of (p,q) analytic functions.
In approximation theory Polozii worked mainly with the aim of developing effective methods to solve boundary value problems which arise in mathematical physics. He work here produced the method of summary representation.*SAU

1970 My Oldest son is born, "Happy Birthday Beau".

DEATHS

1616 Miguel de Cervantes Saavedra died and William Shakespeare both died on this date, the former in Madrid, Spain, the latter in Stratford-on-Avon, England. Which one died ﬁrst? This is not a trick question; they died several days apart. All you need to solve it is some knowledge of the calendar. *VFR (Curiously, Shakespeare was also born on this date in 1564. If you see April 26th, that is date of his baptism.)

1839 The Very Reverend James Wood (14 December 1760 – 23 April 1839) was a mathematician, Dean of Ely and Master of St John's College, Cambridge.
Wood was born in Holcombe where his father ran an evening school and taught his son the elements of arithmetic and algebra. From Bury Grammar School he proceeded to St John's College, Cambridge in 1778, graduating as senior wrangler in 1782. On graduating he became a fellow of the college and in his long tenure there produced several successful academic textbooks for students of mathematics. (The Elements of Algebra (1795); The Principles of Mechanics (1796); The Elements of Optics (1798))
Wood remained for sixty years at St. John's, serving as both President (1802–1815) and Master (1815–1839); on his death in 1839 he was interred in the college chapel and bequeathed his extensive library to the college, comprising almost 4,500 printed books on classics, history, mathematics, theology and travel, dating from the 17th to the 19th centuries.[3]
Wood was also ordained as a priest in 1787 and served as Dean of Ely from 1820 until his death.{He was succeeded by another eminent mathematician, George Peacock)*Wik

1922 Laroy S. Starrett (25 Apr, 1836-23 Apr 1922) was an American inventor and manufacturer who held over 100 patents, many for fine measurement tools, including the micrometer screw guage (patented 29 Jul 1890) that is familiar to present-day machinists and physics lab workers. His first patent (23 May 1865) was for a meat chopper, which he had manufactured for him, but marketed it himself. This product was successful, and his next patents for shoe studs and hooks provided enough income to establish his own factory. He began making a combination square. This was a try-square with a head that could be moved and clamped at any position along the blade, which he patented 26 Feb 1879. He added products including rules, surface guages, and other small tools. His business became the world's largest in his specialty. When he died, it had over five acres of production space, and 1,000 workers. *TIS The company is still making quality instruments today. I've owned a few fine Starrett micrometers and other gauging equipment in my days.

1930 Henry Ernest Dudeney, (10 April 1857–23 April 1930) England's greatest puzzlist. He was unusually skilled at geometrical dissections, cutting a polygon into the smallest number of pieces that can be refitted to make a different type of polygon. He was also the first to apply digital roots, a term he coined, to recreational mathematics. *VFR
In April 1930, Dudeney died of throat cancer in Lewes, where he and his wife had moved in 1914 after a period of separation to rekindle their marriage. Alice Dudeney survived him by fourteen years and died November 21, 1945, after a stroke. Both are buried in the Lewes town cemetery. Their grave is marked by a copy of an 18th century Sussex sandstone obelisk, which Alice had copied after Ernest's death to serve as their mutual tombstone.(would love a photo if anyone is in that area)
For samples of his puzzles, the Amazon Kindle edition is free.

1960 Max von Laue (9 Oct 1879, 23 Apr 1960 at age 80)German physicist who was a recipient of the Nobel Prize for Physics in 1914 for his discovery of the diffraction of X-rays in crystals. This enabled scientists to study the structure of crystals and hence marked the origin of solid-state physics, an important field in the development of modern electronics. *TIS

Norio Ohga, otherwise spelled Norio Oga (January 29, 1930 – April 23, 2011), was the former president and chairman of Sony Corporation, credited with spurring the development of the compact disc as a commercially viable audio format.

He insisted that a CD should hold 75 minutes of music, sufficient for the entire Beethoven's Ninth Symphony, which accounts for the designed 4.8-inch diameter. Having had a career as an opera singer before he joined the company in the 1950s, he remained a music connoisseur, and recognized the importance of improved sound quality made possible by the CD. Sony issued the world's first CD in 1982, and in Japan, within five years the format overtook LP record sales. He rose through the company to become its chief executive in 1989, always pursuing improvements to quality and appealing design, and led Sony's expansion from hardware to software to entertainment including music, films and video games.

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

Monday 22 April 2024

Heron's Cube Root Method

As is my nature, I love to roam through old math journals, and recently I found a 1920 article on a beautiful hand calculation for cube roots by Heron from his long lost Metrica. Until nearly the end of the Nineteenth Century, Math Historians were writing that the ancient Greeks could not calculate square roots (they seldom even mentioned the thought of a cube root) because, "We can't find the evidence so they didn't know it." The discovery of Heron's Metrica in 1897 provided the evidence with several examples of square roots, and one single example of a Cube root.

This is a pretty accurate cube root for 2000 years ago, accurate to the third decimal place, and an error in the root of a little less than 0.0013 that's the total error. If you read it carefully, a question that has plagued all who have studied it. What is the origin of the 100 added to 180 to get 280? Some early researchers thought it was the original number, 100, and dismissed Heron's method as not very accurate for many numbers. But this article provides an explanation that is, in the words of the author, closer than seven digit logarithms for large numbers (he didn't define).

After doing a few by hand, I "cheated" and built a spreadsheet to compute them and give the error. But if you try a few by hand, you will agree that it is way ahead of that thing they call the cube root algorithm in YouTube videos. So I'll let You in on the secret, (but don't tell the other kids!).

Heron's method begins with the surrounding roots of the number. For the 100 example he used, the lower root bound is 4, and the upper is 5, that is, the number 100 is bound by 64 and 125. Now we need to determine what Heron called the excess, and the deficit. The excess is 5^3 - 100 or 25. The deficit is 100 - 4^3 = 36.

Then we form the numerator of the fractional part of the solution, multiply the deficit by the upper root bound, 5, producing 180.

Now for the denominator, we take the 180 from the numerator, and add the product of the excess times the lower bound, producing the unexplained 100. To get our final root estimate, we add the lower root, 4 to this fraction. My calculator gives 4.642857 but for the actual cube root of 100 it gives 4.6415888.. That's a total error of a smidge more than 00126. If you cube the approximation you get 100.08199.

I ran a spreadsheet for numbers from 10 to 330 and notice a few quirks I believe. The error seems greater when the deficit is more than the excess. Which actually makes 100 look worse than average. If you try 90 instead, where the excess and deficit are 35 and 26, almost the opposite of 100 you get much greater accuracy. For 90, the true value is 4.481404747, and the estimate is 4.481481481, and the total error is 0.0000076734. The cube of the estimate is 90.0046 (What we country folk call Pretty Dang Good!)

It also seems that as the numbers get bigger, the errors get slightly smaller, and I can believe the 7 digit logarithm quote.

So if you are driving down the road and factoring the license plate number in front of you is trivial (Oh, Come On, you know you all do that!) try taking the cube root in you head. Fun on the Highway .... (I missed what turn off??????)

ADDEDUM, With a Hat Tip to Keith Raskin;

So if you only read really old journals, you may miss something really important. Case in point, the 2017 Bulletin of Parnas Mathematical Society, which is in Brazil (Brasil) It

The Bulletin has an extension of Heron's root taking method for any odd root. I've only just begun playing with it, but I'll add the method here, and my results for 5th roots.

The system is much the same, but with a power used in the multiplication of the roots and the excess and deficiency, and the first new task is to find out the power k we want to use in those products. The key is to let the root you seek, n, be equal to 2k+1; so for the fifth root, since 2k+1 = 5 gives us k=2, we want to square the lower and upper bounds of the root when we multiply.

Example Find the fifth root of 100 (which your calculator will tell you is 2.51188... ) .

The bounding roots are 2 and 3. The deficit is 100 - 2⁵ = 68, and the excess is 3⁵- 100 = 143. Now we proceed as before, except the numerator will be 3² times the deficit, 3² x 68 = 612.

Now for the denominator we find the product we add to the numerator, which is the square of the lower root bound, 2, times the excess, 143, 2² x 143 = 572. So the fractional part of our estimate will be 612 / (612 + 572) ... which is 153/296, or as a decimal, 0.51689. Adding our lower root bound we get 2.051689. An error of 0.00500.

On This Day in Math - April 22

It can be of no practical use to know
that π is irrational,
but if we can know,
it surely would be intolerable not to know.

~ Edward Titchmarsh

The 112th day of the year; 112 is a practical number (aka panarithmetic numbers), any smaller number can be formed with distinct divisors of 112. Student's might explore the patterns of such numbers.

112 is the side of the square that can be tiled with the the fewest
number of distinct integer-sided squares, discovered by A. J. W. Duijvestijn in 1976

112 is the only 3-digit number such that its factorial raised to the sum of its digits and increased by one is prime. I.e., 112!^(1+1+2)+1 is prime.

112 = 11 + 13 + 17 + 19 + 23 + 29 (sum of consecutive primes) and
= 1x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 (sum of consecutive oblong or pronic numbers)

112 = 7+9+11+13+15+17 +19+21

Douglas W Boone noted that "_Every_ odd multiple of eight greater than 56 is the sum of eight consecutive positive even numbers. (The smallest sum of eight consecutive positive even numbers is 72 = 2+4+6+8+10+12+14+16.) Allowing zero and negative numbers, every odd multiple of eight, period, is the sum of eight consecutive even numbers. The _even_ multiples of eight (that is, multiples of sixteen) are the sum of eight consecutive _odd_ numbers."

EVENTS

1056, the supernova in the Crab nebula was last seen by the naked eye. The creation of the Crab Nebula corresponds to the bright SN 1054 supernova that was independently recorded by Indian, Arabic, Chinese and Japanese astronomers in 1054 AD. The Crab Nebula itself was first observed in 1731 by John Bevis. The nebula was independently rediscovered in 1758 by Charles Messier as he was observing a bright comet. Messier catalogued it as the first entry in his catalogue of comet-like objects. The Earl of Rosse observed the nebula at Birr Castle in 1848, and referred to the object as the Crab Nebula because a drawing he made of it looked like a crab.*Wik

???? In the century and a half between 1725 and 1875, the French fought and won a certain battle on 22 April of one year, and 4382 days later, also on 22 April, they gained another victory. The sum of the digits of the years is 40. Find the years of the battles. For a solution see Ball’s Mathematical Recreations and Essays, 11th edition, p. 27. *VFR (or see this blog)

1715 (O.S.) A total solar eclipse was observed in England from Cornwall in the south-west to Lincolnshire and Norfolk in the east,the first total solar eclipse visible in London for 500 years. This eclipse is known as Halley's Eclipse, after Edmund Halley (1656–1742) who predicted this eclipse to within 4 minutes accuracy. Halley observed the eclipse from London where the city of London enjoyed 3 minutes 33 seconds of totality. He also drew a predictive map showing the path of totality across England. The original map was about 30 km off the observed eclipse path. After the eclipse, he corrected the eclipse path, and added the path and description of the 1724 total solar eclipse.Note: Great Britain didn't adopt the Gregorian calendar until 1752, so the date was considered 22 April 1715. (Under the modern calendar this would be May 3.) *Wik… The Royal Society reports: Edmund Halley, a Fellow of the Royal Society, is most famous for his work on the orbits of comets, predicting when the one that now bears his name would be seen; however, his interests were more widespread. In 1715 the first total solar eclipse for 500 years took place over England and Wales. Halley, a talented mathematician, realized that such an event would generate a general curiosity and requested that the ‘curious’ across the country should observe ‘what they could’ and make a record of the time and duration of the eclipse. At the time, there were only two universities in England and their astronomy professors did not have much luck in observing the event: ‘the Reverend Mr Cotes at Cambridge had the misfortune to be oppressed by too much company’ and ‘Dr John Keill by reason of clouds, saw nothing distinctly at Oxford but the end’. The event did indeed capture the imagination of the nation and the timings collected allowed Halley to work out the shape of the eclipse shadow and the speed at which it passed over the Earth (29 miles per minute).

1906 First American automobiles meet the first American speed bump. In March of 1906, residents of Chatham Borough, New Jersey had begun construction of a speed control device, crosswalks that were five Inches high, constructed of flagstones and cobblestones. Their creation was a plan to slow down the "very fast pace" (10-15 miles per hour) of the new motor carriages that have begone to take over the roads of the center of town. On "April 22, 1906 with great fanfare and many spectators. Bystanders set up seating and vendors sold hot dogs and pop corn to serve the growing group of onlookers. The next day local newspapers reported on the wreckage and carnage from the newly discovered speed reducers." Here is the article from the New York Times on April 23:

There were several persons in the machine, and when the heavy rubber tires struck the elevation there was a palpitation of the machinery and the car shot up several feet in the air. Goggles, hats, a monkey wrench, sidecombs, hairpins and other articles flew in all directions. The crowd gave a cheer and decided the borough’s plan was effective. The ‘bumps' installed by the borough officials of the village of Chatham to check the speed of automobiles through the village had their first test yesterday, and proved a decided success.

1937 "The Law of Anomalous numbers" is read before the American Philosophical Society. This paper described the mathematical idea that is now more commonly called Benford's Law. The paper seems to be available online at the time of this writing.

1939 Frederic Joliot and his group publish their work on the secondary neutrons released in nuclear fission. This was the first demonstration that a chain reaction is indeed possible. Joliot was one of the scientists mentioned in Albert Einstein's letter to President Roosevelt as one of the leading scientists on the course to chain reactions. *Atomic Heritage Foundation

He was a French physicist and husband of Irène Joliot-Curie, with whom he was jointly awarded the Nobel Prize in Chemistry in 1935 ...

1964 The New York Worlds Fair opened in Flushing Meadows, Queens, NY on this day. One technological innovation presented at the fair was the Olivetti Programma 101, one of the first commercial programmable calculators (*The Old Calculator Web Museum "It appears that the Mathatronics Mathatron calculator preceeded [sic] the Programma 101 to market). 40,000 units were sold; 90% of them in the United States where the sale price was $3,200 (increasing to about $3,500 in 1968.)

About 10 Programma 101 were sold to NASA and used to plan the Apollo 11 landing on the Moon.

"By Apollo 11 we had a desktop computer, sort of, kind of, called an Olivetti Programma 101. It was a kind of supercalculator. It was probably a foot and a half square, and about maybe eight inches tall. It would add, subtract, multiply, and divide, but it would remember a sequence of these things, and it would record that sequence on a magnetic card, a magnetic strip that was about a foot long and two inches wide. So you could write a sequence, a programming sequence, and load it in there, and the if you would – the Lunar Module high-gain antenna was not very smart, it didn't know where Earth was. [...] We would have to run four separate programs on this Programma 101 [...]"

— David W. Whittle, 2006

The P101 is mentioned as part of the system used by the US Air Force to compute coordinates for ground-directed bombing of B-52 Stratofortress targets during the Vietnam War. *Wik

In 1970, the first nationwide Earth Day was celebrated in the U.S. as an environmental awareness event celebrated by millions of Americans with marches, educational programs, and rallies. (A local Earth Day celebration had occurred on 21 Mar 1970, in San Francisco, Cal.). Later the same year, President Nixon created the Environmental Protection Agency, or EPA, on 2 Dec 1970 to address America's severe pollution problem. Its mission is to safeguard the nation's water, air and soil from pollution. The agency conducts research, sets standards, monitors activities and helps to enforce environmental protection laws*TIS This stamp was issued in honor of the first celebration of #EarthDay , which took place in 1970.

2012 A rare daytime meteor was seen and heard streaking over northern Nevada and parts of California on Sunday, just after the peak of an annual meteor shower.
Observers in the Reno-Sparks area of Nevada reported seeing a fireball at about 8 a.m. local time, accompanied or followed by a thunderous clap that experts said could have been a sonic boom from the meteor or the sound of it breaking up high over the Earth. While meteors visible at night typically range in size from a pebble to a grain of sand, a meteor large enough to be seen during daylight hours would presumably be as big as a baseball or softball.*Reuters US

A meteor in the sky above Reno, Nevada on April 22, 2012. Image credit: Lisa Warren

Bill Cooke of the Meteoroid Environments Office at NASA’s Marshall Space Flight Center in Huntsville, Ala., estimates the object was about the size of a minivan, weighed in at around 154,300 pounds (70 metric tons) and at the time of disintegration released energy equivalent to a 5-kiloton explosion. *NASA

BIRTHS

1592 Wilhelm Shickard (22 April 1592 – 24 October 1635) He invented and built a working model of the ﬁrst modern mechanical calculator. *VFR

Schickard's machine could perform basic arithmetic operations on integer inputs. His letters to Kepler explain the application of his "calculating clock" to the computation of astronomical tables.
In 1935 while researching a book on Kepler, a scholar found a letter from Schickard and a sketch of his calculator, but did not immediately recognize thedesigns or their great importance. Another twenty years passed before the book's editor, Franz Hammer, found additional drawings and instructions for Schickard's second machine and released them to the scientific community in 1955.A professor at Schickard's old university, Tübingen, reconstructed thecalculator based upon Schickard's original plans; it is still on display there today.
He was a friend of Kepler and did copperplate engravings for Kepler's Harmonice Mundi. He built the first calculating machine in 1623, but it was destroyed in a fire in the workshop in 1624.

Original drawing taken from F. Seck (Editor) 'Wilhelm Schickard 1592–1635, Astronom, Geograph, Orientalist, Erfinder der Rechenmaschine', Tübingen, 1978

1724 Immanuel Kant (22 April 1724 – 12 February 1804) in Konigsberg, Germany. German philosopher, trained as a mathematician and physicist, who published his General History of Nature and theory of the Heavens in 1755. This physical view of the universe contained three anticipations of importance to astronomers. 1) He made the nebula hypothesis ahead of Laplace. 2) He described the Milky Way as a lens-shaped collection of stars that represented only one of many "island universes," later shown by Herschel. 3) He suggested that friction from tides slowed the rotation of the earth, which was confirmed a century later. In 1770 he became a professor of mathematics, but turned to metaphysics and logic in 1797, the field in which he is best known. *TIS

1807 Luigi Palmieri (April 22, 1807 – September 9, 1896) was an Italian physicist and meteorologist. He was famous for his scientific studies of the eruptions of Mount Vesuvius, for his researches on earthquakes and meteorological phenomena and for improving the seismographer of the time. Using a modified Peltier electrometer, he also carried out research in the field of atmospheric electricity. Other scientific contributions included the development of a modified Morse telegraph, and improvements to the anemometer and pluviometer. *Wik

1811 Ludwig Otto Hesse (22 April 1811 in Königsberg, Prussia (now Kaliningrad, Russia)- 4 Aug 1874 in Munich, Germany)Hesse worked on the development of the theory algebraic functions and the theory of invariants. He is remembered particularly for introducing the Hessian (matrix)determinant. *SAU The Hessian matrix is a square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables.*Wik

1816 The French general, Charles Denis Sauter Bourbaki was born. There is a statue of him in Nancy, France, where Jean Dieudonn´e once taught. The polycephalic mathematician Nicolas Bourbaki was named after him. See Joong Fang, Bourbaki, Paideia Press, 1970, p. 24.*VFR

1830 Thomas Archer Hirst FRS (22 April 1830 – 16 February 1892) was a 19th century mathematician, specialising in geometry. He was awarded the Royal Society's Royal Medal in 1883.Hirst was a projective geometer in the style of Poncelet and Steiner. He was not an adherent of the algebraic geometry approach of Cayley and Sylvester, despite being a personal friend of theirs. His specialty was Cremona transformations.*Wik

1884 David Enskog (April 22, 1884, Västra Ämtervik, Sunne – June 1, 1947,Stockholm) was a Swedish mathematical physicist. Enskog helped develop the kinetic theory of gases by extending the Maxwell–Boltzmann equations.*Wik

1887 Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr. He was a member of the Danish national football team for the 1908 Summer Olympics, where he won a silver medal.

A collaboration with Göttingen-based Edmund Landau resulted in the Bohr–Landau theorem, regarding the distribution of zeroes in zeta functions.

Bohr worked in mathematical analysis, founding the field of almost periodic functions, and worked with the Cambridge mathematician G. H. Hardy.

Bohr was also an excellent football player. He had a long playing career with Akademisk Boldklub, making his debut as a 16-year-old in 1903. During the 1905 season he played alongside his brother Niels, who was a goalkeeper. Harald was selected to play for the Danish national team in the 1908 Summer Olympics, where football was an official event for the first time. The opening match of the 1908 Olympic tournament was Denmark's first official international football match. Bohr scored two goals as Denmark beat the French "B" team 9–0. In the next match, the semi-final, Bohr played in a 17–1 win against France, which remains an Olympic record. Denmark faced hosts Great Britain in the final, but lost 2–0, and Bohr won a silver medal. His popularity as a footballer was such that when he defended his doctoral thesis the audience was reported as having more football fans than mathematicians.

Danish football team at the 1908 Olympic games. Bohr is in the top row, 2nd from left.

1891 Sir Harold Jeffreys (22 Apr 1891, 18 Mar 1989 at age 97)English astronomer, geophysicist and mathematician who had diverse scientific interests. In astronomy he proposed models for the structures of the outer planets, and studied the origin of the solar system. He calculated the surface temperatures of gas at less than -100°C, contradicting then accepted views of red-hot temperatures, but Jeffreys was shown to be correct when direct observations were made. In geophysics he researched the circulation of the atmosphere and earthquakes. Analyzing earthquake waves (1926), he became the first to claim that the core of the Earth is molten fluid. Jeffreys also contributed to the general theory of dynamics, aerodynamics, relativity theory and plant ecology.*TIS

1903 Taro Morishima (22 April 1903 in Wakayama, Japan - 8 Aug 1989 in Tokyo, Japan) a Japanese mathematician specializing in algebra who attended University of Tokyo in Japan. Morishima published at least thirteen papers, including his work on Fermat's Last Theorem, and a collected works volume published in 1990 after his death. He also corresponded several times with American mathematician H. S. Vandiver.
Morishima's Theorem on FLT:
Let m be a prime number not exceeding 31. Let p be prime, and let x, y, z be integers such that x^p + y^p + z^p = 0. Assume that p does not divide the product xyz. Then, p² must divide m^{p − 1}-1. *Wik

1904 J(ulius) Robert Oppenheimer was a U.S. theoretical physicist and science administrator, noted as director of the Los Alamos laboratory during development of the atomic bomb (1943-45) and as director of the Institute for Advanced Study, Princeton (1947-66). Accusations as to his loyalty and reliability as a security risk led to a government hearing that resulted the loss of his security clearance and of his position as adviser to the highest echelons of the U.S. government. The case became a cause célèbre in the world of science because of its implications concerning political and moral issues relating to the role of scientists in government. *TIS

1910 Norman Earl Steenrod (April 22, 1910 – October 14, 1971) was a preeminent mathematician most widely known for his contributions to the field of algebraic topology. He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled Universal homology groups. He held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He died in Princeton.
Thanks to Lefschetz and others, the cup product structure of cohomology was understood by the early 1940s. Steenrod was able to define operations from one cohomology group to another (the so-called Steenrod squares) that generalized the cup product. The additional structure made cohomology a finer invariant. The Steenrod cohomology operations form a (non-commutative) algebra under composition, known as the Steenrod algebra.
His book The Topology of Fiber Bundles is a standard reference. In collaboration with Samuel Eilenberg, he was a founder of the axiomatic approach to homology theory. *Wik

1929 Sir Michael Francis Atiyah, OM, FRS, FRSE (22 April 1929, 11 January 2019) was a British mathematician working in geometry.

was awarded the Fields Medal in 1966 for his work in developing K-theory, a generalized Lefschetz fixed-point theorem and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly with Isadore Singer in 2004. Atiyah received a knighthood in 1983 and the Order of Merit in 1992. He also served as president of the Royal Society (1990-95). *TIS *Wik

A Möbius band is the simplest non-trivial example of a vector bundle.

1946 Paul Charles William Davies, AM (22 April 1946, ) is an English physicist, writer and broadcaster, currently a professor at Arizona State University as well as the Director of BEYOND: Center for Fundamental Concepts in Science. He has held previous academic appointments at the University of Cambridge, University of London, University of Newcastle upon Tyne, University of Adelaide and Macquarie University. His research interests are in the fields of cosmology, quantum field theory, and astrobiology. He has proposed that a one-way trip to Mars could be a viable option.*Wik

DEATHS

1945 Wilhelm Cauer (June 24, 1900 – April 22, 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prior to his work, electronic filter design used techniques which accurately predicted filter behaviour only under unrealistic conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter. *Wik
By the end of World War II, he was, like millions of less-distinguished countrymen and -women, merely a person in the way of a terrible conflagration.
Cauer succeeded in evacuating his family west, where the American and not the Soviet army would overtake it — but for reasons unclear he then returned himself to Berlin. His son Emil remembered the sad result.

The last time I saw my father was two days before the American Forces occupied the small town of Witzenhausen in Hesse, about 30 km from Gottingen. We children were staying there with relatives in order to protect us from air raids. Because rail travel was already impossible, my father was using a bicycle. Military Police was patrolling the streets stopping people and checking their documents. By that time, all men over 16 were forbidden to leave towns without a permit, and on the mere suspicion of being deserters, many were hung summarily in the market places. Given this atmosphere of terror and the terrible outrages which Germans had inflicted on the peoples of the Soviet Union, I passionately tried to persuade my father to hide rather than return to Berlin, since it was understandable that the Red Army would take its revenge. But he decided to go back, perhaps out of solidarity with his colleagues still in Berlin, or just due to his sense of duty, or out of sheer determination to carry out what he had decided to do.
Seven months after the ending of that war, my mother succeeded in reaching Berlin and found the ruins of our house in a southern suburb of the city. None of the neighbors knew about my father’s fate. But someone gave identification papers to my mother which were found in a garden of the neighborhood. The track led to a mass grave with eight bodies where my mother could identify her husband and another man who used to live in our house. By April 22, 1945, the Red Army had crossed the city limits of Berlin at several points. Although he was a civilian and not a member of the Nazi Party, my father and other civilians were executed by soldiers of the Red Army. The people who witnessed the executions were taken into Soviet captivity, and it was not possible to obtain details of the exact circumstances of my father’s death.

*ExecutedToday.com

1948 Herbert William Richmond (17 July 1863 Tottenham, England – 22 April 1948 Cambridge, England) was a mathematician who studied the Cremona–Richmond configuration. He was elected a Fellow of the Royal Society in 1911. T
The Cremona–Richmond configuration is a configuration of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles.*Wik

1989 Emilio Gino Segrè (1 Feb 1905; 22 Apr 1989) was an Italian-born American physicist who was co-winner, with Owen Chamberlain of the United States, of the Nobel Prize for Physics in 1959 for the discovery of the antiproton, an antiparticle having the same mass as a proton but opposite in electrical charge. He also created atoms of the man-made new element technetium (1937) and astatine (1940). Technetium occupied a hitherto unfilled space in the body of the Periodic Table, and was the first man-made element not found in nature. Astatine exists naturally only in exceedly small quantities because as a decay product of larger atoms, and having a half-life of only a few days, it quickly disappears by radioactively decay to become atoms of another element.*TIS

L to R Rasetti, Fermi,Segre.

2001 John Frank Allen, FRS FRSE (May 5, 1908 – April 22, 2001) was a Canadian-born physicist. codiscovered the superfluidity of liquid helium near absolute zero temperature. Working at the Royal Society Mond Laboratory in Cambridge, with Don Misener he discovered (1930's) that below 2.17 kelvin temperature, liquid helium could flow through very small capillaries with practically zero viscosity. Independently, P. L. Kapitza in Moscow produced similar results at about the same time. Their two articles were published together in the 8 Jan 1938 issue of the journal Nature. Superfluidity is a visible manifestation resulting from the quantum mechanics of Bose- Einstein condensation. By 1945, research in Moscow delved into the microscopic aspect, which Allen did not pursue.*TIS

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2002 Victor Frederick Weisskopf (September 19, 1908 – April 22, 2002) was an Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli and Niels Bohr. During World War II he worked at Los Alamos on the Manhattan Project to develop the atomic bomb, and later campaigned against the proliferation of nuclear weapons.
His brilliance in physics led to work with the great physicists exploring the atom, especially Niels Bohr, who mentored Weisskopf at his institute in Copenhagen. By the late 1930s, he realized that, as a Jew, he needed to get out of Europe. Bohr helped him find a position in the U.S.
In the 1930s and 1940s, 'Viki', as everyone called him, made major contributions to the development of quantum theory, especially in the area of Quantum Electrodynamics.[3] One of his few regrets was that his insecurity about his mathematical abilities may have cost him a Nobel prize when he did not publish results (which turned out to be correct) about what is now known as the Lamb shift. *Wik

2008 Derek Thomas "Tom" Whiteside FBA (23 July 1932 – 22 April 2008) was a British historian of mathematics. He was the foremost authority on the work of Isaac Newton and editor of The Mathematical Papers of Isaac Newton. From 1987 to his retirement in 1999, he was the Professor of History of Mathematics and Exact Sciences at Cambridge University. *Wik