Thursday, 21 June 2018

On This Day in Math - June 21

I had this rare privilege of being able 
to pursue in my adult life, 
what had been my childhood dream.
~Andrew Wiles

The 172nd day of the year; seventeen 2's followed by two 17's is prime.*Prime Curios
222222222222222221717 is prime

\( 172 = \pi(1+7+2) * p_{(1*7*2)} \). It is the only known number (up to 10^8) with this property.
pi(n) is the number of primes less than or equal to n, and pn is the nth prime.


1667 Louis XIV, The Sun King of France, attends the ceremony of inauguration of the Observatoire de Paris, the oldest working observatory in the world. *Amir D. Aczel, Pendulum, pg 66

1669 Christopher Wren gives first proof that the hyperboloid of one sheet (Wren uses the term Hyperbolic Cylindroid.) is doubly ruled in the Philosophical Transactions of the Royal Society. The only three doubly ruled surfaces are the plane, the hyperboloid of one sheet, and the hyperbolic paraboloid. Wren includes an image of the hyperboloid of one sheet that may be the earliest ever in print. In a footnote in Boyer's History of History of Analytic Geometry he notes that there is a figure in Kepler's Stereometria which looks like it might be this shape. (It is interesting that in his work on the geometry of a barrel, Kepler gives an approximation formula for the volume of a barrel that is exact for the hyperboloid of one sheet.)
The invention of the telescope and efforts to reduce distortion in the lenses led to suggestions of hyperbolic lenses, and Wren's paper pointed out "an application thereof for grinding hyperbolical glasses." Newton had applied the knowledge that the hyperboloid of one sheet was doubly ruled in his notes in 1666 when he demonstrated how to turn the shape on a lathe holding the cutting tool obliquely to the axis of rotation.
The image of Newton's method below is from a paper by Professor Rickey on the net.
*Wik, *VFR,

1798 Cavendish reads a paper to the Royal Society of London describing experiments to measure the density of the earth, and hence its weight, with results that it is 5.48 times the density of water. (the figures seem to include at least one calculating error) *Philosophical Transactions, 1798, Part II, pgs 469-526

1808 on 30 June, Humphry Davy announced he had separated the element boron. However, working independently, French chemist, Joseph Louis Gay-Lussac had announced* the same accomplishment nine days ealier, on 21 Jun 1808*TIS

In 1886, the foundation stone of the Tower Bridge in London, England was laid (over a time capsule) by the Prince of Wales. The need to cross the River Thames at this point had become increasingly urgent for many years, and finally the necessary Act was passed in 1885. The bridge, designed by Mr. Wolfe Barry, CB, was completed at a cost of about £1,000,000. To permit the passage of tall ships between the towers, two bascule spans, each of 100-ft length, are raised. The side spans to the towers are of the more familiar suspension type. Pedestrians can traverse a high-level footway nearly at the top of the towers, even when the bridge is raised. It was officially opened 30 Jun 1894, by the Prince of Wales, later Edward VII, on behalf of Queen *TIS

In 1893, the first Ferris wheel premiered at Chicago's Columbian Exposition, America's third world's fair. It was invented by George Washington Ferris, a Pittsburgh bridge builder, for the purpose of creating an attraction like the Eiffel Tower in Paris. Each of the 36 cars carried 60 passengers, making a full passenger load of 150 tons. Ferris didn't use rigid spokes: instead, he used a web of taut cables, like a bicycle wheel. Supported by two 140 foot steel towers, its 45 foot axle was the largest single piece of forged steel at the time in the world. The highest point of the wheel was 264 feet. The wheel and cars weighed 2100 tons, with another 2200 tons of associated levers and machinery. Ferris died just four years later, at the age of only 38. *TIS

1929 Kazimierz Kuratowski (1896–1980) at a meeting of the Warsaw Section of the Polish Mathemat­ical Society, announced that a graph is planar iff it does not contain a subgraph homeomorphic to either K–5, the complete graph on 5 points, or K–3–3, the complete bipartite graph on two sets of three points. See HM 12, 258, for a discussion of the early history of this theorem which is now the most cited result in graph theory. *VFR (See June 18) 
 "A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three)." *Wik 
 (in more simple, but less exact terms,  "it can be drawn in such a way that no edges cross each other."  The well-known recreational problem of connecting three houses to three utilities is not possible to draw because it is K3,3 (below).  The utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any of the gas/water/electric lines/pipes pass over any other. (1913 Dudeney: first publication of Gas, Water and Electricity Problem. according to David Singmaster, Gardner says 1917)

1948 the first stored-program computer, the Small-Scale Experimental Machine, SSEM, ran its first program. Written by Professor Tom Kilburn, it took 52 minutes to run. The tiny experimental computer had no keyboard or printer, but it successfully tested a memory system developed at Manchester University in England. The system, based on a cathode-ray tube, could store programs. Previous electronic computers had to be rewired to execute each new problem. The Manchester computer proved theories set forth by John von Neumann in a report that proposed modifications to ENIAC, the electronic computer built at the University of Pennsylvania in the mid-1940s. The report also proposed the use of binary instead of digital numbers. *TIS

1976 Kenneth Appel and Wolfgang Haken announced that with the aid of a computer that they had proved the four color problem. Because of the use of the computer the solution was not quickly accepted by all, but today most mathematicians accept the proof as correct. However, no simple proof is known as yet. *VFR  {A really nice article on the four color theorem and its history}
In 1963 Donald B. Gillies had found three new primes. When the primes were confirmed the UIUC Math dept (which has a postal branch) used this cancellation stamp on all mail from roughly 1964 - 1976. When Appel and Haken proved the four color theorem ("Four Colors Suffice") a new stamp was created. Trivia question : how far away from Gillies did Appel live in Urbana Illinois ??
Answer : He lived 3 houses away. *Wik
*Wik courtesy of Chris Caldwell

1993   Andrew Wiles  begins the three days of lectures leading to a solution of Taniyama-Shimura conjecture, and completing the proof of Fermat’s last theorem.. See (June 23)

2011  On non-leap years (until 2039), this day marks the summer solstice in the northern hemisphere and the winter solstice in the southern hemisphere, and this is the day of the year with the longest hours of daylight in the northern hemisphere and the shortest in the southern hemisphere.*Wik


1710 James Short (June 10 {June 21 NS), 1710, Edinburgh, Scot. -  June 14, 1768, London, Eng) British optician and astronomer who produced the first truly
parabolic and elliptic (hence nearly distortionless) mirrors for reflecting telescopes. During his working life of over 35 years, Short made about 1,360 instruments - not only for customers in Britain but also for export: one is still preserved in Leningrad, another at Uppsala and several in America. Short was principal British collator and computer of the Transit of Venus observations made throughout the world on 6th June 1761. His instruments travelled on Endeavour with Captain Cook to observe the next Transit of Venus on 3rd June 1769, but Short died before this event took place.

 1781 Siméon-Denis Poisson ( 21 June 1781 – 25 April 1840) French mathematician known for his work on definite integrals, advances in Fourier series, electromagnetic theory, and probability. The Poisson distribution (1837) describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times. His works included applications to electricity and magnetism, and astronomy. He is also known for the Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity.*TIS   Libri wrote of him: “His only passion has been science: he lived and is dead for it.” *VFR

1852 Eduard Weyr (1852-1903) He and his brother, Emil Weyr (1848–1894) were the leading members of the Austrian geometrical school. They worked in descriptive geometry, projective geometry, and then became interested in algebraic and synthetic methods. Eduard found a canonical form for matrices that deserves to be better known (American Mathematical Monthly, December 1999). *VFR

1863 Maximilian Franz Joseph Cornelius Wolf was a German astronomer who founded and directed the Königstuhl Observatory. He used wide-field photography to study the Milky Way and used statistical treatment of star counts to prove the existence of clouds of dark matter. He was among the first astronomers to show that the spiral nebulae have absorption spectra typical of stars and thus differ from gaseous nebulae. His most important contribution was the introduction of photography to discover hundreds of asteroids, the first of which he named Brucia in honor of the donor of his 16-inch double telescope, Catherine Wolfe Bruce.*TIS

1918 Tibor Szele worked in group theory. *VFR  Hungarian mathematician, working in combinatorics and abstract algebra. After graduating at the Debrecen University, he became a researcher at the Szeged University in 1946, then he went back to Debrecen University in 1948 where he became full professor in 1952. He worked especially in the theory of Abelian groups and ring theory. He generalized Hajós's theorem. He founded the Hungarian school of algebra. *Wik

1954 David Ríos Insua (born June 21, 1964 in Madrid) is a Spanish mathematician, and son and disciple of Sixto Ríos, father of Spanish Statistics. He is currently also the youngest Fellow of the Spanish Royal Academy of Sciences (de la Real Academia de Ciencias Exactas, Físicas y Naturales, RAC),[1] which he joined in 2008.[2][3] He received a PhD in Computational Sciences at the University of Leeds. He is Full Professor of the Statistics and Operations Research Department at Rey Juan Carlos University (URJC),[4] and he has been Vice-dean of New Technologies and International Relationships at URJC (2002–2009). He has worked in fields such as Bayesian inference in neuronal networks, MCMC methods in decision analysis, Bayesian robustness or adversarial risk analysis. He has also worked in applied areas such as Electronic Democracy,[5] reservoirs management, counterterrorism model and many others. He is married and has two daughters. Wik


1874 Anders Jonas Ångström was a Swedish physicist whose pioneering use of spectroscopy is recognised in the name of the angstrom, a unit of length equal to 10-10 metre. In 1853, he studied the spectrum of hydrogen for which Balmer derived a formula. He announced in 1862 that analysis of the solar spectrum showed that hydrogen is present in the Sun's atmosphere. In 1867 he was the first to examine the spectrum of aurora borealis (northern lights). He published his extensive research on the solar spectrum in Recherches sur le spectre solaire (1868), with detailed measurements of more than 1000 spectral lines. He also published works on thermal theory and carried out geomagnetical measurements in different places around Sweden.*TIS

1913  Gaston Tarry was a French combinatorialist whose best-known work is a method for solving mazes.*SAU  He also was able to confirm Leonhard Euler's conjecture that no 6×6 Graeco-Latin square was possible. 
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint sets A and B of n integers each, such that:
\sum_{a\in A} a^i = \sum_{b\in B} b^i
for each integer power  i from 1 to a given k.
For example, a solution with n = 6 and k = 5 is the two sets { 0, 5, 6, 16, 17, 22 } and { 1, 2, 10, 12, 20, 21 }, because:
01 + 51 + 61 + 161 + 171 + 221 = 11 + 21 + 101 + 121 + 201 + 211
02 + 52 + 62 + 162 + 172 + 222 = 12 + 22 + 102 + 122 + 202 + 212
03 + 53 + 63 + 163 + 173 + 223 = 13 + 23 + 103 + 123 + 203 + 213
04 + 54 + 64 + 164 + 174 + 224 = 14 + 24 + 104 + 124 + 204 + 214
05 + 55 + 65 + 165 + 175 + 225 = 15 + 25 + 105 + 125 + 205 + 215.
This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Escott, who studied it in the early 1910s.

1940 Wolfgang Döblin, known in France as Vincent Doblin, (17 March 1915 – 21 June 1940) was a German-French mathematician. Wolfgang was the son of the Jewish-German novelist, Alfred Döblin. His family escaped from Nazi Germany to France where he became a citizen. Studying probability theory at the Institute Henri Poincaré under Fréchet, he quickly made a name for himself as a gifted theorist. He became a doctor at age 23. Drafted in November 1938, after refusing to be exempted of military service, he had to stay in the active Army when World War II broke out in 1939, and was quartered at Givet, in the Ardennes, as a telephone operator. There, he wrote down his latest work on the Chapman-Kolmogorov equation, and sent this as a "pli cacheté" (sealed envelope) to the French Academy of Sciences. His company, sent to the sector of the Saare on the ligne Maginot in April 1940, was caught in the German attack in the Ardennes in May, withdrew to the Vosges, and capitulated on June 22, 1940. On June 21, Doeblin had committed suicide in Housseras (a small village near to Epinal), at the moment where German troops came in sight of the place. In his last moments, he burned his mathematical notes.
The sealed envelope was opened in 2000, revealing that Döblin was ahead of his time in the development of the theory of Markov processes. In recognition of his results, Itō's lemma is now referred to as the Itō–Doeblin Theorem.
His life was recently the subject of a movie by Agnes Handwerk and Harrie Willems, A Mathematician Rediscovered. *Wik

1948 D'Arcy Thompson graduated from Cambridge University in Zoology. He was a appointed Professor of Biology at Dundee and later Professor of Natural History at St Andrews. He combined skills in a way that made him unique. He was a Greek scholar, a naturalist and a mathematician. He was the first biomathematician. He became an honorary member of the EMS in 1933. *SAU

1957  Johannes Stark German physicist who won the 1919 Nobel Prize for Physics for his discovery in 1913 that an electric field would cause splitting of the lines in the spectrum of light emitted by a luminous substance; the phenomenon is called the Stark effect. *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

Wednesday, 20 June 2018

On This Day in Math - June 20

the enormous success of mathematics in the natural
sciences is something bordering on the mysterious and ...
there is no natural explanation for it.
—Eugene Wigner

The 171st day of the year; 171 has the same number of digits in Roman numerals as its cube.

\( 10^{171 } - 171 \)is prime

Google calculator gives 171! = infinity. (close enough in many cases)

171 is the last year-day that is both a triangular number and a palindrome. *Ben Vitale


1686 Halley Writes to Newton that Hooke has protested his "discovery" of the inverse square law should be noted in Principia. Newton will respond On July 14, 1686, with a peace offering; "And now having sincerely told you the case between Mr Hooke and me, I hope I shall be free for the future from the prejudice of his letters. I have considered how best to compose the present dispute, and I think it may be done by the inclosed scholium to the fourth proposition." This scholium was "The inverse law of gravity holds in all the celestial motions, as was discovered also independently by my countrymen Wren, Hooke and Halley."

1688 Newton, in a letter to Edmund Halley, again expresses his exasperation with carping critics. [Thanks to Howard Eves]*VFR

1788; Washington Writes to Nicholas Pike to Thank him for a copy of his "A New and Complete System of Arithmetic" , published in 1786 by Nicholas Pike, a Newburyport schoolmaster. In his letter, sent June 20, 1788, from Mount Vernon, Washington writes: "The handsome manner in which that Work is printed and the elegant manner in which it is bound, are pleasing proofs of the progress which the Arts are making in this Country. Washington's letter to Pike also commended him on his accomplishments and the importance of his work.
Pike had written to  Washington on March 25,1786 requesting permission to dedciate the book to Washington. On June 20 of 1786, Washington had replied that, "I must therefore beg leave to decline the honour which you would do me, as I have before done in two or three cases of a similar kind."

1808 Poisson submitted his first paper on the stability of the planetary system, one day before his twenty-seventh birthday. *VFR

1831 János Bolyai's pioneering work, The Absolutely True Science of Space, was published in 1832. This important work was published as an appendix to the first volume of his father,Farkas Bolyai's Tentamen , but its off-print had already been ready the previous year, in April 1831. The latter was the version which, together with a letter, was sent to Gauss by Farkas Bolyai on the 20th of June 1831. Gauss got the letter but János's work was lost on the way. On the 16th of January 1832 Farkas sent the Appendix to his friend 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''.
After twenty-three years of silence, Gauss replied to his ``old, unforgettable friend'' on the 6th of March 1832. One of his 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

1877 Georg Cantor, in a letter to Dedekind, announced a proof that the points inside a square are in one-to-one correspondence with those on a line segment. Three years earlier, Cantor had intimated that this was clearly impossible. *VFR

1908 Count Zeppelin made his first flight in his fourth new airship at Friedrichshafen, Germany. The Luftschiff LZ4 had its first flight 20 Jun 1908. Its first extended flight (12 hours) was taken to Switzerland 1 Jul 1908. At the beginning of August, it embarked on an extended flight which had taken it among other places to Basel, Straussberg, and many of the major cities of southern Germany. While moored at Echterdingen on 5 Aug 1908, it was torn from the mast by high winds and destroyed. As interest in the Zeppelins ran high in German, the incident was felt as a national disaster. Spontaneous donations resulted in approximately 5.5 million Marks and made it possible for Zeppelin to continue his work. *TIS


1775 Jacques Frédéric Français (20 June 1775 in Saverne, Bas-Rhin, France - 9 March 1833 in Metz, France) In September 1813 Français published a work in which he gave a geometric representation of complex numbers with interesting applications. This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to François Français. Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians. This changed after Français' paper since a vigorous discussion between Français, Argand and Servois took place in Gergonne's Journal. In this argument Français and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra. *SAU

1838 Theodor Reye (20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) worked in Geometry and Projective Geometry.*SAU

1873 Alfred Loewy born.(20 June 1873 in Rawitsch, Germany (now Rawicz, Poznań, Poland) - 25 Jan 1935 in Freiburg im Breisgau, Germany) He worked in group theory and differential equations. *VFR

1940 Leonard Susskind ( June(20ish 1940)(The professor's real birthday seems difficult to determine; perhaps only known to him and his parents, perhaps only to his parents) is the Felix Bloch Professor of Theoretical Physics at Stanford University, and Director of the Stanford Institute for Theoretical Physics. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology. He is a member of the National Academy of Sciences, and the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics,[6] and a distinguished professor of the Korea Institute for Advanced Study.
Susskind is widely regarded as one of the fathers of string theory, having, with Yoichiro Nambu and Holger Bech Nielsen, independently introduced the idea that particles could in fact be states of excitation of a relativistic string. He was the first to introduce the idea of the string theory landscape in 2003. *Wik


1800 Abraham Kästner (27 September 1719 – 20 June 1800) was a German mathematician who compiled encyclopaedias and wrote text-books. He taught Gauss. *SAU

1807 Ferdinand Berthoud (19 March 1727 – 20 June 1807) Outstanding Swiss horologist and author of extensive treatises on timekeeping who became involved in the attempt to solve the problem of determining longitude at sea. His major achievement was his further development of an accurate and practical marine clock, or chronometer. (Such an instrument had previously been constructed in expensive and delicate prototypes by Pierre Leroy of France and John Harrison of England.) He made his first chronometer in 1754, which was sent for trial in 1761. Berthoud's improvements to the chronometer have been largely retained in present-day designs. *TIS

1861 Sir Frederick Gowland (Hoppy)Hopkins OM PRS (20 June 1861 – 16 May 1947) was an English biochemist who was awarded the Nobel Prize in Physiology or Medicine in 1929, with Christiaan Eijkman, for the discovery of vitamins. He also discovered the amino acid tryptophan, in 1901. He was President of the Royal Society from 1930 to 1935. His Cambridge students included neurochemistry pioneer Judah Hirsch Quastel and pioneer embryologist Joseph Needham.
During his life, in addition to the Nobel Prize, Hopkins was awarded the Royal Medal of the Royal Society in 1918 and the Copley Medal of the Royal Society in 1926. Other significant honours were his election in 1905 to fellowship in the Royal Society, Great Britain's most prestigious scientific organisation; his knighthood by King George V in 1925; and the award in 1935 of the Order of Merit, Great Britain's most exclusive civilian honour. From 1930 -1935 he served as president of the Royal Society and in 1933 served as President of the British Association for the Advancement of Science. *Wik

1865 Sir John William Lubbock, (London, England, 26 March 1803 - Downe, Kent, England, 20 June 1865 )English astronomer and mathematician. He made a special study of tides and of the lunar theory and developed a method for calculating the orbits of comets and planets. In mathematics he applied the theory of probability to life insurance problems. He was a strong proponent of Continental mathematics and astronomy.
Lubbock, third Baron Lubbock, was born into a London banking family. After attending Eton, he moved to Trinity College, Cambridge, where he became a student of William Whewell.(it was at the request of Lubbock that Whewell created the term "biometry".) He excelled in mathematics and traveled to France and Italy to deepen his knowledge of the works of Pierre-Simon de Laplace and Joseph Lagrange. Entering his father’s banking firm as a junior partner, he devoted his free time to science.
Lubbock strongly supported Lord Brougham’s Society for the Diffusion of Useful Knowledge [SDUK], which produced scientific and technical works designed for the working class. His articles on tides for the Society’s publications resulted in a book, *An Elementary Treatise on the Tides, in 1839. *Biographical Encyclopedia of Astronomers

1966 Georges (Henri) Lemaître was a Belgian astronomer and cosmologist, born in Charleroi, Belgium. He was also a civil engineer, army officer, and ordained priest. He did research on cosmic rays and the three-body problem. Lemaître formulated (1927) the modern big-bang theory. He reasoned that if the universe was expanding now, then the further you go in the past, the universe’s contents must have been closer together. He envisioned that at some point in the distant past, all the matter in the universe was in an exceedingly dense state, crushed into a single object he called the "primeval super-atom" which exploded, with all its constituent parts rushing away. This theory was later developed by Gamow and others.*TIS

2003 I. Bernard Cohen (1 March 1914 – 20 June 2003) was the Victor S. Thomas Professor of the history of science at Harvard University and the author of many books on the history of science and, in particular, Isaac Newton.
Cohen was the first American to receive a Ph.D. in history of science, was a Harvard undergraduate ('37) and then a Ph.D. student and protégé of George Sarton who was the founder of Isis and the History of Science Society. Cohen taught at Harvard from 1942 until his death, and his tenure was marked by the development of Harvard's program in the history of science. *Wik

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

Tuesday, 19 June 2018

On This Day in Math - June 19

The more I see of men, the better I like my dog.
Blaise Pascal (over 350 years before Carrie Underwood)

The 170th day of the year; the start of a record-breaking run of consecutive integers (170-176) with an odd number of prime factors.

170 is the smallest number that can be written as the sum of the squares of 2 distinct primes, where each of these primes is the square of a prime added to another prime (170 = (22 + 3)2 + (32 + 2)2).
*Prime Curios

170 is the largest integer for which its factorial can be stored in double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306. (For 171! it returns "infinity".)

170 is the smallest number n for which phi(n)(the number of integers relatively prime to 170=64=82) and sigma(n) (the sum of the divisors of 170=324=182) are both square.


In 240 BC, Eratosthenes, a Greek astronomer and mathematician, estimated the circumference of the earth. As the director of the great library of Alexandria, he read in a papyrus book that in Syene, approaching noon on the summer solstice, the longest day of the year, shadows of temple columns grew shorter. At noon, they were gone. The sun was directly overhead. However, a stick in Alexandria, far to the north, could cast a pronounced shadow. Thus, he realized that the surface of the Earth could not be flat. It must be curved. Not only that, but the greater the curvature, the greater the difference in the shadow lengths. By measurement on the ground and application of geometry, he calculated the circumference of the earth. *TIS

325 The early Christian church opened the council of Nicaea, which decided the rules for computing the date of Easter: The first Sunday after the first full moon on or after the vernal equinox *VFR

1934 Jerzy Neyman's paper before the Royal Statistical Society entitled "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. This paper was the one first presenting the concept of a "confidence interval" (interval estimate). *David Bee

1934 1st motion picture of the solar surface was made using the McMath-Hulbert Spectroheliokinematograph :
K8MHO is the club radio station for the McMath-Hulbert Astronomical Society. The station is housed in the McGregor administration building of the McMath-Hulbert Solar Observatory which at one time was the second largest solar observatory in the world. The station is currently manned by Tom Hagen, NE9Y, and Dave Benham, K8TRF. Members of this radio club have a mutual interest in astronomy, ham radio and the preservation of the McMath-Hulbert Solar Observatory.
The McMath-Hulbert Solar Observatory was founded in 1929 by Francis McMath, his son Robert McMath and Henry Hulbert, neighbors who just had a mutual interest in astronomy. The first tower at this site was built with a 16 foot dome in 1930 and originally had a 10.5” equatorial telescope. As they gained more interest in observing the sun, this building became more exclusively devoted to solar observing. On June 19, 1934, they released the first ever motion picture film of the surface of the sun. *QRZ.Com with hat tip to David Dickinson ‏@Astroguyz

In 1963, Soviet cosmonaut Valentina Tereshkova returned to Earth after spending nearly three days as the first woman in space. She had been interested in parachute jumping when she was young, and that expertise was one of the reasons she was picked for the cosmonaut program. She became the first person to be recruited without experience as a test pilot. On 16 Jun 1963, Tereshkova was launched into space aboard Vostok 6, and became the first woman to travel in space. Her radio name was "Chaika," Russian for "seagull." Her flight made 48 orbits of Earth. Tereshkova never made a second trip into space. She became an important member of the Communist Party and a representative of the Soviet government.*TIS


1623 Blaise Pascal ( 19 June 1623 – 19 August 1662) born in Ferrand, Auvergne, France.  He laid the foundation for the modern theory of probabilities. In hydrodynamics he formulated what came to be known as Pascal's law of pressure, and invented the syringe and hydraulic press. Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years (1642-45). The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Schickard had manufactured one in 1624. He died at the young age of 39 having been sickly and physically weak through life. Autopsy showed he had been born with a deformed skull.*TIS

1669 Leonty Magnitsky (June 9, 1669, Ostashkov – October 19, 1739, Moscow) was a Russian teacher who wrote the first guide to mathematics published in Russia.*SAU

1771 Joseph Gergonne born. (19 June 1771 Nancy, France—4 May 1859 Montpellier, France)  He came under the influence of Gaspard Monge, the Director of the new École Polytechnique in Paris. In 1810, in response to difficulties he encountered in trying to publish his work, Gergonne founded his own mathematics journal, officially named the Annales de mathématiques pures et appliquées but generally referred to as the Annales de Gergonne. The most common subject of articles in his journal was geometry, Gergonne's specialty. Over a period of 22 years, the Annales de Gergonne published about 200 articles by Gergonne himself, and other articles by many distinguished mathematicians, including Poncelet, Servois, Bobillier, Steiner, Plücker, Chasles, Brianchon, Dupin, Lamé, even Galois.
Gergonne was appointed to the chair of astronomy at the University of Montpellier in 1816. In 1830, he was appointed Rector of the University of Montpellier, at which time he ceased publishing his journal. He retired in 1844.
Gergonne was the first mathematician to employ the word polar. In a series of papers beginning in 1810, he discovered the principle of duality in projective geometry, by noticing that every theorem in the plane connecting points and lines corresponds to another theorem in which points and lines are interchanged, provided that the theorem embodied no metrical notions. In 1816, he devised an elegant solution to the problem of Apollonius: find a circle which touches three given circles.
In 1813, Gergonne wrote the prize-winning essay for the Bordeaux Academy, Methods of synthesis and analysis in mathematics, unpublished to this day and known only via a summary. The essay is very revealing of Gergonne's philosophical ideas. He called for the abandonment of the words analysis and synthesis, claiming they lacked clear meanings. Surprisingly for a geometer, he suggested that algebra is more important than geometry, at a time when algebra consisted almost entirely of the elementary algebra of the real field. He predicted that one day quasi-mechanical methods would be used to discover new results.
In 1815, Gergonne wrote the first paper on the optimal design of experiments for polynomial regression. According to S. M. Stigler, Gergonne is the pioneer of optimal design as well as response surface methodology.

1846 Antonio Abetti (19 Jun 1846, 20 Feb 1928 at age 81) Italian astronomer who was an authority on minor planets. At first a civil engineer, he became an astronomer at the University of Padua (1868-93), with an interest in positional astronomy and made many observations of small planets, comets and star occultations. In 1874, Abetti went to Muddapur, Bengal, to observe the transit of Venus across the sun's disk where his use of a spectroscope was the first use of this kind. Later, he became director at the Arcetri Observatory and Professor of astronomy at the University of Florence (1894-1921). The observatory had been founded by G. B. Donati in 1872, and Abetti equipped it with a new telescope that he had built in the workshops at Padua. He was active after retirement, until his death, and was followed by his son Giorgio.*TIS

1851 Silvanus P. Thomson (19 June 1851 – 12 June 1916) In 1910 he published Calculus Made Easy, which was published anonymously until after his death in 1916. It is still in print. *VFR He was a noted physicist and engineer, and a celebrated teacher and writer on electricity and magnetism. He also wrote popular biographies of Faraday and Lord Kelvin. At his death he was professor at City and Guilds Technical College at Finsbury (London). Thompson’s particular gift was in his ability to communicate difficult scientific concepts in a clear and interesting manner. He attended and lectured at the Royal Institution giving the Christmas lectures in 1896 on Light, Visible and Invisible with an account of Röntgen Light. He was an impressive lecturer and the radiologist AE Barclay said that: “None who heard him could forget the vividness of the word-pictures he placed before them.”


1504 Bernhard Walther (1430 – June 19, 1504) was a German merchant, humanist and astronomer based in Nuremberg, Germany.
Walther was born in Memmingen, and was a man of large means, which he devoted to scientific pursuits. When Regiomontanus settled in Nuremberg in 1471, they worked in collaboration to build an observatory and a printing press. After the death of Regiomontanus in 1476 at Rome, Walther bought his instruments, after Hans von Dorn, commissioned by the Hungarian king, had tried in vain about it with the council of Nuremberg. Thenceforward, he continued the observation of planets till his death in Nuremberg. His house, purchased in 1509 by Albrecht Dürer, is nowadays a museum

1945 Stefan Mazurkiewicz (September 25, 1888 in Warsaw, then Russian Empire – June 19, 1945, Grodzisk Mazowiecki, Poland)one of the founders of Fundamenta Mathematicae, died.

*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, 18 June 2018

On This Day in Math - June 18

I began to understand that pure mathematics was more than a collection of random tools mainly fashioned for use in the Cambridge treatment of natural philosophy.
Andrew Forsyth

The 169th day of the year; 169 is the smallest square which is prime when rotated 180o (691)  What is the next one?

And from Jim Wilder, 169 is the reverse of 961. The same is true of their square roots... √169=13 and √961=31
or stated another way, 169 = 132 and in reverse order 312 = 961

An interesting loop sequence within Pi. If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, *Pi Search page

169 is the only year day which is both the difference of consecutive cubes, and a square: \(8^3-7^3 =169=13^2\)

The first successful dissection of a square into smaller squares was of a square with 169 units on a side. 1907-1914 S. Loyd published The Patch Quilt Puzzle. A square quilt made of 169 square patches of the same size is to be divided into the smallest number of square pieces by cutting along lattice lines. The answer, which is unique, is composed of 11 squares with sides 1,1,2,2,2,3,3,4,6,6,7 within a square of 13. It is neither perfect nor simple. Gardner states that this problem first appeared in 1907 in a puzzle magazine edited by Sam Loyd. David Singmaster lists it as first appearing in 1914 in Cyclopedia by Loyd but credits Loyd with publishing Our Puzzle Magazine in 1907 - 08. This puzzle also appeared in a publication by Henry Dudeney as Mrs Perkins Quilt. Problem 173 in Amusements in Mathematics. 1917


1558 Robert Recorde’s will was admitted to probate, after he died in prison. He introduced the equals sign in The Whetstone of Witte (1557) with the words: “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lenghte, thus:  because noe .2. thynges, can be moare equalle.” “Gemowe” (think Gemini )is an old French work meaning “twin.”. *VFR  When they are asked what they would use if this was not available, it seems difficult for students to imagine a different symbol.  

Image from Wikipedia.

1584 Jacob Christmann appointed professor of Hebrew at Heidelberg. In 1595 he defended the view that the circle could only be approximately squared. *VFR

1864 Lewis Carroll finally decided to write up Alice’s Adventures in Wonderland. [Stuart Dodgson Collingwook, The Life and Letters of Lewis Carroll (1898), p. 96] 

1908  Alan Archibald Campbell Swinton took the first x-ray images in Britain in January 1896 and by a year later the medical professions were bringing him surgical cases for analysis. But "on this day he predicted exactly how another magic box would work, in a letter to Nature. He called it ‘Distant Electric Vision’, but we know it now as television." *Keith Moore,

 1928, aviator Amelia Earhart became the first woman to fly across the Atlantic Ocean. She had accepted the invitation of the American pilots Wilmer Stultz (1900-29) and Louis Gordon to join them on the transatlantic flight. The crossing from Newfoundland to Wales took about 21 hours. Amelia Earhart went on to establish herself as a respected role model, tirelessly demonstrating that young women were as capable as men in succeeding in their chosen vocations. In 1935 she crossed the Atlantic solo in record time: 13 hr 30 min.  *TIS

1983 Sally Ride, astrophysicist, becomes the first American woman in space. The Soviets were ahead by twenty years and two days.*VFR


1799 William Lassell (18 June 1799 – 5 October 1880) was a wealthy amateur English astronomer. He set up an observatory at Starfield, near Liverpool. England, He built his own 24" diameter telescope, and devised steam-driven equipment for grinding an polishing the speculum metal mirror. This telescope was the first of its size to be mounted "equitorially" to allow easy tracking of the stars. He discovered Triton, a moon of Neptune, and Ariel and Umbriel, satellites of Uranus. Later, Lassell built a 48" diameter telescope with th same design and took it to Malta for observations with clearer skies.*TIS

1818 Pietro Angelo Secchi (18 Jun 1818, 26 Feb 1878 at age 59) Italian Jesuit priest and astrophysicist, who made the first survey of the spectra of over 4000 stars and suggested that stars be classified according to their spectral type. He studied the planets, especially Jupiter, which he discovered was composed of gasses. Secchi studied the dark lines which join the two hemispheres of Mars; he called them canals as if they where the works of living beings. (These studies were later continued by Schiaparelli.) Beyond astronomy, his interests ranged from archaeology to geodesy, from geophysics to meteorology. He also invented a meteorograph, an automated device for recording barometric pressure, temperature, wind direction and velocity, and rainfall.*TIS

1858 Andrew Russell Forsyth (18 June 1858, Glasgow – 2 June 1942, South Kensington) studied at Liverpool College and was tutored by Richard Pendlebury before entering Trinity College, Cambridge, graduating senior wrangler in 1881. He was elected a fellow of Trinity and then appointed to the chair of mathematics at the University of Liverpool at the age of 24. He returned to Cambridge as a lecturer in 1884 and became Sadleirian Professor of Pure Mathematics in 1895. *Wik 
In 1893 he published Theory of functions of a complex variable which had such an impact at Cambridge that function theory dominated there for many years. Whittaker writes... that this text:-
... had a greater influence on British mathematics than any work since Newton's Principia.
However the reputation of the book outside Britain was not high. In fact this is not surprising since the whole thrust of the book was to bring the great advances of Continental mathematics to Cambridge which Forsyth rightly saw as living in the past. He was well equipped to undertake this task for he traveled widely and, being a good linguist, was able to appreciate the advances made by authors writing in French and German.
On Cayley's death Forsyth was appointed to his chair in 1895 becoming the Sadleirian professor of Pure Mathematics. However his preference for technical mastery rather than rigorous analysis meant that he failed to inspire future pure mathematicians. In fact one would have to say that Forsyth was unlucky, for although he saw the importance of Continental mathematics, at the same time his greatest strengths lay in his ability to handle complex formulae. He therefore excelled at precisely the style of mathematics which he himself campaigned successfully to replace at Cambridge.

1884  Charles Weatherburn  (18 June, 1884 in Australia - 1974 in Australia) worked on vector analysis and differential geometry.*SAU

1884 Frieda Nugel (18 June 1884 in Cottbus, Brandenburg, Germany- 6 Nov 1966 in Bad Godesberg, Bonn, Germany) was a German mathematician who was one of the first women to receive a doctorate in Germany *SAU

1913  Oswald Teichmüller's (June 18, 1913 – September 11, 1943)  main contribution is in the area of geometric function theory.*SAU

1926 Allan Rex Sandage  (June 18, 1926 – November 13, 2010)  U.S. astronomer who (with Thomas A. Matthews) discovered, in 1960, the first optical identification of a quasi-stellar radio source (quasar), a starlike object that is a strong emitter of radio waves. Although a strange source of radio emission, in visible light, it looked like a faint star. Yet this object was emitting more intense radio waves and ultraviolet radiation than a typical star. He is best known for determining the first reasonably accurate value for the Hubble constant and the age of the universe.*TIS & Wik


1818 George Baron (?? , June 18, 1818) was a mathematician who emigrated from Northumberland, England to Hallowell, Maine in the United States, thereafter moving to New York. He was the first superintendent and mathematics professor at what would become the United States Military Academy in 1801 and the founder and editor-in-chief of the Mathematical Correspondent, which was the first American "specialized scientific journal" and the first American mathematics journal, first published May 1, 1804.
Baron was first offered the position at the fledgling academy at West Point, New York by the newly elected United States President Thomas Jefferson's Secretary of War Henry Dearborn, a friend of Baron's who had lived near him in Maine. After agreeing upon salary and perks, instruction began on September 21, 1801 employing the use of Charles Hutton's A Course in Mathematics and a blackboard, the first recorded use of the latter in America. In October, there was a disagreement between Baron and one of the cadets, Joseph Gardner Swift. Swift was called upon to apologize and was reprimanded for the language he employed against Baron, but went on to become the Military Academy's first graduate, and later a Brigadier General. For a variety of reasons, Baron was court-martialled in December, and Major Jonathan Williams became the supervisor and Captain William Amherst Barron became the instructor of mathematics.
Baron became a teacher of mathematics in New York City, there joining the Theistical Society of New York, a deist group led by Elihu Palmer that came to public attention in the course of a pamphlet war between supporters of United States Vice President Aaron Burr and supporters of then United States Senator from New York DeWitt Clinton. *Wik
Baron may have been one of the earliest users of a blackboard in the US as, "use of the blackboard was a favorite method of Baron." *Edward S Holden, The Centenial of the US Military Academy at West Point, New York

1922 Jacobus Cornelius Kapteyn, (January 19, 1851, Barneveld, Gelderland – June 18, 1922) Dutch astronomer who used photography and statistical methods in determining the motions and spatial distribution of stars. Such work was the first major step after the works of William and John Herschel. He tried to solve the questions of space density of stars as a function of distance from the sun, and the distribution of starts according to brightness per unit volume. Some of his results had lasting value, but some were superceded because he had failed to account for the interstellar absorption. In studies using proper motion to determine stellar distances, he discovered stellar motions are not random, as previously thought, but that stars move in two "star streams" (1904). He introduced absolute magnitude and colour index as standard concepts.*TIS

1935 Alexander von Brill (20 September 1842 – 18 June 1935) died. He worked on algebraic geometry and the theory of algebraic func­tions.  Born in Darmstadt, Hesse, he attended University of Giessen where he earned his doctorate under supervision of Alfred Clebsch. He held a chair at the University of Tübingen, where Max Planck was among his students.*Wik

1980 Kazimierz Kuratowski  (February 2, 1896 – June 18, 1980) He worked in the area of topology and set theory. He is best known for his theorem giving a necessary and sufficient condition for a graph to be planar.*SAU   
Kuratowski's theorem:  "A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three)."  (in simpler, but less exact terms,  it can be drawn in such a way that no edges cross each other."  The well-known recreational problem of connecting three houses to three utilities is not possible to draw because it is K3,3 (below).  The utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any of the gas/water/electric lines/pipes pass over any other. (1913 Dudeney: first publication of Gas, Water and Electricity Problem. according to David Singmaster, Gardner says 1917)  (see June 21)

Kuratowski proved his theorem in 1930. Forty years later the dedication of Frank Harary’s classic Graph Theory was:
Who gave K5 and K3,3
To those who thought planarity
Was nothing but topology.
(In fact three other almost simultaneous discoveries of the theorem are recorded: Orrin Frink and Paul Althaus Smith; Lev Semenovich Pontrjagin; and Karl Menger!)  With thanks to *theoremoftheday@ theoremoftheday

*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