On This Day in Math - June 26

  

When you measure what you are speaking about and express it in numbers,
you know something about it,
but when you cannot express it in numbers
your knowledge about is of a meagre and unsatisfactory kind.

William Thomson, Lord Kelvin


The 177th day of the year; there are 177 graphs with seven edges.  *What's So Special About This Number.  (only 79 of these are connected graphs)

• 177 is the smallest magic constant for a 3 x 3 prime magic square

An old idea that was new to me, from John Golden@mathhombre and here.  Use two Simple Squares to make a  third, and I just did one with magic sum of 177.
2   3   1
1   2   3
3   1   2 

And 
53  61  57 
61  57  53
57  53  61

Now add the numbers in same location

55  64  58
62  59  56
60  54  63  

77 is also a Leyland number,  expressible as a^b + b^a. both greater than one . using 2 and 7 in this case.  . The numbers are named for British mathematician Paul Layland from Oxford University.  There are only nine year days that are Leyland numbers.  Only one of those nine is prime.

More Math Facts for each day here

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EVENTS

1424 Of the 20 total eclipses to visit the Orkneys and Shetland Islands in the period 1 - 3000AD it was the 13th longest in the whole of the UK at 3 minutes 56 seconds it was surpassed in Orkney by those of 364, 885, 1185, 1433, 2681. The eclipse track traveled across Denmark, Germany, Poland, Ukraine, Moldavia, and the Black Sea. (ref. SW-UK eclipses) *NSEC

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1614   The first lottery of significance in the new world was held on this date by the Virginia Company.  The first Great Prize was 4,500 Crowns. *JN Kane, Famous First Facts (I have seen the date of this lottery also given as 1612)

Lotteries held a prominent place in the early history of America, including an important role in financing the establishment of the first English colonies. The first such lottery, in 1612, raised 29,000 pounds for the Virginia Company. Lotteries were frequently used in colonial-era America to finance public works projects such as paving streets, constructing wharves, even building churches. In the 18th century, lotteries were used to finance construction of buildings at Harvard and Yale. George Washington sponsored a lottery in 1768 to build a road across the Blue Ridge Mountains, but it was unsuccessful.  (*NATIONAL GAMBLING IMPACT STUDY COMMISSION)  

The article mentioned above included this statement, "A more far-reaching development was the advent of on-line computerized vending. The first major innovation from this was a daily numbers game, modeled on the illegal numbers games historically present in all major American cities. The advantages to the player of this new, legal game included the ability to choose his own "lucky" number, thereby giving him a greater sense of participation (even if his actual odds of winning remained unaffected by his choices) "  I wonder how many statisticians would disagree with the comment in parentheses?

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1765 Benjamin Franklin writes to Peter Collinson about numerous topics including Accounts of Spouts and Whirlwinds, and a comment on his earlier kite experiments; but includes, "I am endeavouring to answer Dr. Parsons’s Request relating to the Indian Names of the Cardinal Numbers." *franklinpapers

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1794,The Battle of Fleurus, was the most significant battle of the Flanders Campaign in the Low Countries during the French Revolutionary Wars. It was the first battle in history that incorporated aerial reconnaissance and observation of an enemy force. This was provided by a French helium balloon, l'Entreprenant, operated by a small crew.  Almost all comments about the balloons effectiveness by both sides dismissed its influence as non-existent.  

An alternative view is expressed at the science Photo Library where they write, "The National Convention became interested in using the balloon to observe enemy maneuvers and appointed Coutelle a captain and first officer of the Company of Aeronauts. They ordered him to build balloons to aid the French Revolutionary armies. The balloon L'Entreprenant became the first military reconnaissance aircraft and was first used in the 1794 conflict with Austria where, in the Battle of Fleurus, they gathered information. Moreover, the presence of the balloon had a demoralizing effect on the Austrian troops which improved the likelihood of victory for the French troops."

 l'EntreprenantJourdan at Fleurus with the balloon l'Entreprenant in the background. Painted by Mauzaisse in 1837; on display in the Galerie des Batailles, Versailles.

*Credit SCIENCE SOURCE / SCIENCE PHOTO LIBRARY

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1819, The first US patent for a velocipede, a predecessor of the bicycle, was issued to William K. Clarkson Jr. of New York. Little information remains available, however, because a fire at the Patent Office in 1836 destroyed the patent record, and it was not restored. 

The first known bicycle was shown by the Comte de Sivrac, who in 1791 was seen riding a two-wheel "wooden horse" in the gardens of the Palais Royal in Paris. Called a celerifere, the machine had two rigidly mounted wheels, so that it was incapable of being steered. To change direction, it was necessary to lift, drag, or jump the front wheel to one side.

In 1817, Charles, Baron von Drais, of Sauerbrun, devised a front wheel capable of being steered. He also gave it a padded saddle, and an armrest in front of his body, which assisted him in exerting force against the ground. Granted a patent in 1818, he took his Draisienne to Paris, where it was again patented and acquired the name vélocipède, a term that was to continue in use until about 1869 when the word “bicycle” came into use.

1818 Draisienne 

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1881 The great comet of 1881. Observed on the night of June 25-26 at 1h. 30m. A.M. from a print by Étienne Léopold Trouvelot, a French artist, astronomer and amateur entomologist. He is noted for the unfortunate introduction of the Gypsy Moth into North America. *The New York Public Library Digital Collections

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1896 An early x-ray photograph of Sir William Crookes’s hand, taken with a cathode tube that bears his name, the Crookes Tube. The man taking these pioneering radiographs was the engineer Alan Archibald Campbell Swinton, later a Fellow of the Royal Society. He 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. *Keith Moore, Royal Society Blog

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In 1974, at 8:01 a.m., a package of Wrigley's chewing gum with a bar code printed on it passed over a scanner at the Marsh Supermarket, Troy, Ohio, and became the first product ever logged under the new Universal Product Code (UPC) computerized recognition system. Invented by IBM, and approved for use in 1973, the UPC is a 12-number bar code representing the manufacturer's identity and an assigned product number. Within nanoseconds, this information is read with a laser beam moving at around 10,000 inches per second and transfers it to the store's database computer for price lookup and inventory management*TIS

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In 1984, the National Maritime Museum, of which the Royal Observatory, Greenwich is a part, encouraged people up and down the Line to organise events in order to mark the so-called ‘centenary’ of the Prime Meridian. Although the International Meridian conference took place in October 1884, the Museum designated Tuesday 26 June as ‘Meridian Day’, on the grounds that any outdoor events would be less likely to be affected by the weather.
Commemorative six-inch diameter plastic plaques were offered to any individual who could show that the Meridian passed through the curtilage of their property. Potential claimants were required to write to their regional office of the Ordnance Survey to verify their claim and send this as proof of authenticity to the English Tourist Board who were distributing them. No records of how many were issued can be traced. The locations of just four are known, along with the existence of a fifth.
The National Maritime Museum also arranged for the Enfield Foundry to cast a bronze plaque as a more enduring alternative. At the time, it was stated that they would only be produced if 20 or more orders were received. How many were made is unknown, the Foundry’s records having been destroyed. Only three have been located to date. (If you are aware of one of these locations, please informe me, thanks PB)

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In 2000, the completion of a working draft reference DNA sequence of the human genome was announced at the White House by President Bill Clinton, and representatives from the Human Genome Project (HGP) and the private company Celera Genomics. Clinton stated that even greater discoveries would follow from the working draft. As a draft, it contained some gaps and errors, but represented about 95% of all genes. HGP expected to use it as a scaffold for generating the high-quality reference genome sequence within three years. This provides knowledge to link genes with particular diseases, of the influence of genetics and to help discover new treatments.
*TIS

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BIRTHS

1730 Charles Messier (26 June 1730 – 12 April 1817) French astronomer who discovered 15 comets. He was the first to compile a systematic catalog of "M objects." The Messier Catalogue (1784), containing 103 star clusters, nebulae, and galaxies. (In Messier's time a nebula was a term used to denote any blurry celestial light source.) He established alphanumeric names for the objects (M1, M2, etc.), which notation continues to be used in astronomy today.

The Orion Nebula as drawn by Messier, and which he gave the designation M 42 in his catalog 

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1824 Lord Kelvin (26 June 1824 – 17 December 1907) Born as William Thomson, he became an influential physicist, mathematician and engineer who has been described as a Newton of his era. At Glasgow University, Scotland, he was a professor for over half a century. The name he made for himself was more than just a temperature scale. His activities ranged from being the brains behind the laying of a transatlantic telephone cable, to attempting to calculate the age of the earth from its rate of cooling. In 1892, when raised to the peerage as Baron Kelvin of Largs, he had chosen the name from the Kelvin River, near Glasgow.*TIS

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1878 Leopold Löwenheim (26 June 1878 in Krefeld, Germany (also the birthplace of Max Zorn) – 5 May 1957 in Berlin) was a German mathematician who worked on mathematical logic and is best-known for the Löwenheim-Skolem paradox.*SAU

The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin. Nevertheless, he survived the Second World War, after which he resumed teaching mathematics.

Löwenheim (1915) gave the first proof of what is now known as the Löwenheim–Skolem theorem, often considered the starting point for model theory.

 

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 1915 Paul Joseph Kelly (June 26, 1915 – July 15, 1995) was an American mathematician who worked in geometry and graph theory. Kelly was born in Riverside, California. He earned bachelor's and master's degrees from the University of California, Los Angeles before moving to the University of Wisconsin–Madison for doctoral studies; he earned his Ph.D. in 1942 with a dissertation concerning geometric transformations under the supervision of Stanislaw Ulam.

Kelly is known for posing the reconstruction conjecture with his advisor Ulam, which states that every graph is uniquely determined by the ensemble of subgraphs formed by deleting one vertex in each possible way. He also proved a special case of this conjecture, for trees.

He is the coauthor of three textbooks: Projective geometry and projective metrics (1953, with Herbert Busemann), Geometry and convexity: A study in mathematical methods (1979, with Max L. Weiss), and The non-Euclidean, hyperbolic plane: Its structure and consistency (1981, with Gordon Matthews).

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1918 Yudell Leo Luke (26 June 1918 – 6 May 1983) was an American mathematician who made significant contributions to the Midwest Research Institute, was awarded the N. T. Veatch award for Distinguished Research and Creative Activity in 1975, and appointed as Curator's Professor at the University of Missouri in 1978, a post he held until his death. Luke published eight books and nearly 100 papers in a wide variety of mathematical areas, ranging from aeronautics to approximation theory. By his own estimation, Luke reviewed over 1800 papers and books throughout his career.*SAU

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1936 Gonzalo Pérez Iribarren (b. June 26, 1936, Carmelo – August 27, 1998, Montevideo ) was a Uruguayan mathematician and statistician .

In addition to cultivating literature, painting, and being a great fan of photography, Gonzalo Pérez had a strong vocation for the application of mathematics in the natural sciences. Thus, among his contributions is his pioneering activity in the study of the flows of the great Uruguayan rivers (the Uruguay River and the Negro River ), which gave rise to the discovery of the incidence of the El Niño phenomenon on the climate. of Uruguay.*Wik

He was forced to spend ten years in exile during the right-wing dictatorship beginning in 1973. He returned to Uruguay in 1983 and was the main force in rebuilding the Institute of Mathematics and Statistics and setting up the Basic Sciences Development Programme.*SAU

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 1937 Robert Coleman Richardson (June 26, 1937 – February 19, 2013) was an American physicist who (with Douglas Osheroff and David Lee) was awarded the 1996 Nobel Prize for Physics for their discovery of superfluidity in the isotope helium-3. As helium is reduced in temperature toward almost absolute zero, a strange phase transition occurs, and the helium takes on the form of a superfluid. The atoms had until that point had moved with random speeds and directions. But as a superfluid, the atoms then move in a co-ordinated manner! 

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1969 Andrei Yuryevich Okounkov (June 26, 1969 - ) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at Columbia University. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry." *Wik

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DEATHS

1274 Nasir al-Tusi (born 18 February 1201 in Ṭūs, Khorasan – died on 26 June 1274 in al-Kāżimiyyah district of metropolitan Baghdad), was an Islamic astronomer and mathematician who joined the Mongols who conquered Baghdad. He made important contributions to astronomy and wrote many commentaries on Greek texts.*SAU Among the many wonderful antiquities at the Bodleian Library is a 16th century printing of the 13th century Arabic translation by Nasir al-Din al-Tusi of Euclid's Elements. It was part of the collection donated by Thomas Allen.

Manuscript diagram of a Tusi couple, in ink with annotations, inserted at the page with a printed Tusi couple, in the Library’s copy of Nicholas Copernicus, De revolutionibus, 1543 (Linda Hall Library, photo by Karl Galle)

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1796 David Rittenhouse (April 8, 1732 – June 26, 1796) American astronomer, instrument maker and inventor who was an early observer of the atmosphere of Venus. For observations for the transit of Venus on 3 Jun 1769, he constructed a high precision pendulum clock, an astronomical quadrant, an equal altitude instrument, and an astronomical transit. He was the first one in America to put spider web as cross-hairs in the focus of his telescope. He is generally credited with inventing the vernier compass and possibly the automatic needle lifter. He was professor of astronomy at the University of Pennsylvania. Benjamin Franklin consulted him on various occasions. For Thomas Jefferson he standardized the foot by pendulum measurements in a project to establish a decimal system of weights and measures.*TIS

Diagram from David Rittenhouse's
observations of the 1769 transit of Venus *WIK

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1810 Joseph Montgolfier (26 August 1740 – 26 June 1810) French ballooning pioneer, with his younger brother, Étienne. An initial experiment with a balloon of taffeta filled with hot smoke was given a public demonstration on 5 Jun 1783. This was followed by a flight carrying three animals as passengers on 19 Sep1783, shown in Paris and witnessed by King Louis XVI. On 21 Nov 1783, their balloon carried the first two men on an untethered flight. In the span of one year after releasing their test balloon, the Montgolfier brothers had enabled the first manned balloon flight in the world. *TIS

irst public demonstration in
Annonay, 5 June 1783 *Wik

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 1914 Lyman Spitzer Jr. (June 26, 1914 – March 31, 1997) was American astrophysicist who advanced knowledge of physical processes in interstellar space and pioneered efforts to harness nuclear fusion as a clean energy source. He made major contributions in stellar dynamics and plasma physics. He founded study of the interstellar medium (gas and dust between stars from which new stars are formed). Spitzer studied in detail interstellar dust grains and magnetic fields as well as the motions of star clusters and their evolution. He studied regions of star formation and was among the first to suggest that bright stars in spiral galaxies formed recently. Spitzer was the first person to propose the idea of placing a large telescope in space and was the driving force behind the development of the Hubble Space Telescope. *Tis

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1951 George Udny Yule (18 February 1871 – 26 June 1951) graduated in Engineering from University College London and then studied in Bonn. He worked with Karl Pearson on the statistics of regression and correlation. He took a post with an examinations board before being appointed to a Cambridge fellowship. He is best known for his book: Introduction to the Theory of Statistics.*SAU

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1967 Henry Thomas Herbert Piaggio (2 June 1884–26 June 1967) graduated from Cambridge and then worked at the University of Nottingham. He is best known for his text-book on Differential Equations.("An Elementary Treatise on Differential Equations and their Applications".) *SAU

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1990 Joseph Carl Robnett Licklider (March 11, 1915 – June 26, 1990), known simply as J.C.R. or "Lick" was an American computer scientist, considered one of the most important figures in computer science and general computing history. He is particularly remembered for being one of the first to forsee modern-style interactive computing, and its application to all manner of activities; and also as an Internet pioneer, with an early vision of a world-wide computer network long before it was built. He did much to actually initiate all that through his funding of research which led to a great deal of it, including today's canonical graphical user interface, and the ARPANET, the direct predecessor to the Internet.*Wik

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1997 Robert Wertheimer Frucht (later known as Roberto Frucht) (9 August 1906 – 26 June 1997)] was a German-Chilean mathematician; his research specialty was graph theory and the symmetries of graphs.

Frucht is known for Frucht's theorem, the result that every group can be realized as the group of symmetries of an undirected graph, and for the Frucht graph, one of the five smallest cubic graphs without any nontrivial symmetries. LCF notation, a method for describing cubic Hamiltonian graphs, was named for the initials of Joshua Lederberg, H. S. M. Coxeter, and Frucht, its key developers.

With Coxeter and David L. Powers, Frucht was coauthor of a book on zero-symmetric graphs, the cubic graphs that have exactly one symmetry taking each vertex to each other vertex.

Frucht was elected to the Chilean Academy of Sciences as a corresponding member in 1979. A special issue of the Journal of Graph Theory was published in Frucht's honor in 1982, and another special issue of the journal Scientia, Series A (the journal of the mathematics department of Federico Santa María Technical University) was published in honor of his 80th birthday in 1986.  *Wik

The Frucht graph.

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2011 Jack Warga ( Warsaw , September 20 , 1922 - Boynton Beach, June 26, 2011 ) was an American mathematician , born in Poland . 

Warga was born in Warsaw , the son of a Jewish furrier. After suffering an anti-Semitic racist attack, his father decided enough was enough and in 1938 sent him to study in Brussels where he had family. When he finished secondary studies, he entered the École Polytechnique de Bruxelles , where he could not even finish the first year because in 1940 the Nazis invaded Belgium . The young Warga escaped from Belgium via Paris and Vichy , but since France did not seem like a very safe place for a Jew either, he continued his escape through Spain and Cuba , until he arrived in New York in 1943. The older part of his family died in the holocaust . 

Since arriving in the United States he served in the United States Army as an officer, until the war ended in 1945, when he continued his studies at New York University . In 1950 he obtained his doctorate with a dissertation on number theory supervised by Harold N. Shapiro. In the following years he worked as a mathematician in different naval research organizations, until in 1957 he became the chief mathematician of the research and development division of the aerospace company AVCO in Wilmington (Massachusetts) , a position he maintain until 1966 when he was appointed professor at Northeastern University in Boston .  Warga retired in 1993.

Warga's research work was in the field of applied mathematics . More specifically, his contributions were pioneering in control and optimization theory, in which he introduced the concept of relaxed control .  His book Optimal Control of Differential and Functional Equations (1972) was for many years the standard text on the subject

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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

Nedials, Nedians, and some related triangles and some properties

On the day I wrote my blog about some interesting properties of the medians of a triangle, I received a package of old Mathematics Teacher articles from Dave Renfro. One of the first I looked at was a January 1951 "Mathematical Miscellanea" edited by Phillip S. Jones. The article contained a contribution by John Satterly of the University of Toronto on a type of cevian that he called "Nedians". [My personal choice, since the "med" root is for the middle, would have just been to call them n-dians, but I'm sure that would have had a cultural backlash.]

For those who may not be familiar with the term "cevian", it refers to a segment in a triangle from a vertex to the opposite side (extended if necessary). Angle bisectors, medians, and altitudes are all cevians then, but a perpendicular bisector of a side would not be because it doesn't necessarily pass through a vertex. The name is in honor of Giovanni Ceva and was originated in France in 1888 and has spread from there.

Professor Satterly seems to have created the term "nedians" as a comparison for medians to describe a cevian that cuts the opposite side 1/n th of the way from one vertex to the next. [I shall use the notation 4-nedian for a nedian that cuts 1/4th of the way along the opposite side,] A median would be the 2-nedian.
In the image, triangle ABC has 3-nedians AD, BE, and CF where D is 1/3 of the way from B to C; E is 1/3 of the way from C to A, etc.

The intersections of the three Nedians of a triangle will form another triangle at their three points of intersection, called the nedian triangle. [JKL in the image].

Professor Satterly seems to have discovered several properties of the nedians, and their nedian triangles which I will give here; and then I have come up with several interesting properties of my own about them that I will add to this blog.

It is not too difficult using affine properties of a triangle to verify many of these.
Professor Satterly showed that the sum of the squares of the nedians would be  times the sum of the squares of the sides of the original triangle. Notice that when n=2, this reduces to 3/4, the ratio given and proved for the medians in the previous blog.

Students may wish to explore these properties by creating Sketchpad or Geogebra interactive models to confirm them, and then challenge themselves to prove them. Proving them is easier with a property of affine geometry. Every triangle is affine equivalent to every other triangle in the plane. Under affine transformations areas and lengths may change, but ratios of them are preserved... which means that you can choose any triangle... a right triangle or an equilateral triangle, to find a property about ratios of lengths or areas, and it will apply to any other triangle... and that is a BIG idea to lock away (I am not well schooled in the particulars of affine geometry, so if I have mis-stated that in some way, please advise).

Professor Satterly also stated that the Nedian triangle will have an area of \(\frac{(n-2)^2}{n^2-n+1}\) times the area of ABC. Note that for the median, or 2-nedian, the area diminishes to zero since the three medians intersect in a single point.

Professor Satterly suggested the term "backward nedian triangle" for a case in which the 1/n ratio went in the opposite order (Let D be 1/3 of the way from C to B insted). This can be eliminated if we simply allow any real number for the coefficient. Then the backward 3-nedian is just a 3/2-nedian in the regular order, and it seems that all his properties are still preserved. Notice that the areas of the 3 and 3/2-nedians are equal, but they are not congruent.

I later found that this idea Professor Satterly calls nedians was written on by Edward John Routh,   a famous Mathematical Tripos coach, in 1891 simply using the term cevians.  The idea had appeared earlier as a  question in the Mathematical Tripos of 1878. Routh's Theorem applies to any ratio between the two parts of a triangle side, even if all three sides are divided in different proportions.  

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Exploring these nedians constructions a little more, I came up with a few more properties that were not in the article. For example, the perpendicular distance from the three vertices of the nedian triangle to any side of ABC will equal the altitude of ABC to that same side. [I call these the sub-altitudes.]

Here the altitude is shown in bold red, and the three corresponding sub-altitudes are shown in dotted red. In the 3-nedian shown the distance from J to side B plus the distance from K to side B plus the distance from L to side B will equal the altitude from B to side B. A similar result exists for each altitude of the triangle. In addition, the three sub-altitudes will always partition the altitude in the same way. The shortest sub-altitude will be , while the next longer one will be  and the longest will be .

It is also clear from the last statement that each of the three small triangles at the vertices of A, B, and C will be congruent. Their bases will each be 1/n of a base of the original triangle and their heights are  so each of them is an equal fraction,  of the original area of ABC. By similar reasoning we see that all three of the quadrilaterals will also have the same area.

I also observed that that each nedian is partitioned into three parts whose lengths, in order from the vertex to 

the opposite side, are .

For the 3-nedians in the image, for example, the nedian CF is partitioned so that CJ is 3/7of CF; JL is 3/7 of CF; and LF is 1/7 of LF. A similar partition holds for the other two 3-nedians AD and BE. In a 4-nedian, the partitions would be 4/14, 8/13, and 1/14.

The Nedial Triangle

 I have played around a little more with the concept and come up with a calculation for the area of what I call a nedial triangle, extending the idea of a medial triangle.


Most High School students are introduced to the Medians of a triangle, and it is quite easy to show using basic high school geometry that the triangle whose vertices are the three feet  of the medians has an area equal to 1/4 the area of the original triangle.  In fact, it is pretty easy to show that the original triangle can be dissected into four congruent copies of the medial triangle.



If instead we use the feet of the three nedians, we get a triangle I have called the nedial triangle.  I have worked out that the area of such a triangle for an n-nedian (the cevian that cuts the opposite side 1/n th of the way along the edge) will have an area of
times the area of the original triangle ABC.  Since the median is the 2-Nedian, this gives the correct ratio for the medial triangles, 1/4.

The image below shows the 4-nedial triangle, which has an area of 7/16 the area of ABC.



The area of the three triangles at the vertices of the original triangle will each be
times the area of ABC.

The area of the nedial triangle will grow from a minimum of 1/4 the area of ABC increasing toward a limit of one as n approaches infinity.  This can be confirmed by L'Hopital's rule or simple division of the terms.  The fact that the limit is also one as n approaches one points out that when the base of the nedian is more than half way along the opposite side and the "n" in the nedian must be less than 2 but more than one we are measuring the area of what Satterly called the "backward nedians" which will create a nedial triangle of the same area. n=3/2 gives the exact area of n=3 (although the triangles are NOT congruent).  In general for every k-nedial the area given by the nedial with index equal to ⁿ/_n-1 has the same area.

On This Day in Math - June 25

 

Astronomy was the cradle of the natural sciences and the starting point of geometrical theories.

~Cornelius Lanczos


The 176th day of the year; 176 and its reversal 671 are both divisible by 11. ( Students should confirm that the reverse of  any number that is divisible by 11 will also be divisible by 11.)

176 is a happy number, repeatedly iterating the sum of the squares of the digits will lead to 1, 1² + 7² + 6²= 86, 8² + 6² = 100 and 1² + 0² + 0² = 1

The number 15 can be partitioned in 176 ways. For younger students, imagine all the different ways of making fifteen cents with US coins, 1 cent, 5 cents, and 10 cents.... now imagine there were also coins worth 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, and 15 cents. There would be 176 different collections of coins that would total exactly 15 cents.

176 is a Self number, it can't be written by any other number plus the sum of its digits. 21 for instance, is not a self number because 15+1+5 = 21.

8*20 + 16 = 176 so 176 = 24^2 - 20 ^2  (try this on 192 also ) and it is divisible by 16 with quotient 11, so 15^2 - 7^2 = 176.

EVENTS

1641 John Pell begins the work of expanding Walter Warner's table of anti-logarithms from 10,000 to 100,000 entries. William Warner [ (mathematician, physiologist and philosopher, and last surviving member of the circle of Thomas Hariot; he corresponded with Mersenne in 1639-40] felt he was too old to complete the laborious task he had set for  himself, and offered Pell 40 GBPounds (appx. worth 5,000 pounds today) to complete the tables and make them ready for printing.  *Thomas Harriot's Doctrine of Triangular Numbers, Beery & Stedall, pg 39

Pell's published works were few: the Controversiae pars prima in 1647, An Introduction to Algebra in 1668, and the Table of 10000 square numbers in 1672. These books represent just a fraction of a lifetime of mathematical activity; the rest can be discovered from Pell's correspondence and his unpublished papers. *John Pell (1611-1685) and His Correspondence with Sir Charles Cavendish: The Mental World of an Early Modern Mathematician 2nd Edition, Malcolm & Stedall

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1665 René Descartes died on 11 February 1650 in Stockholm, Sweden, where he had been invited as a teacher for Queen Christina of Sweden. The cause of death was said to be pneumonia—accustomed to working in bed until noon, he may have suffered a detrimental effect on his health due to Christina's demands for early morning study (the lack of sleep could have severely compromised his immune system). Others believe that Descartes may have contracted pneumonia as a result of nursing a French ambassador, Dejion A. Nopeleen, ill with the aforementioned disease, back to health. In his recent book, Der rätselhafte Tod des René Descartes (The Mysterious Death of René Descartes), the German philosopher Theodor Ebert asserts that Descartes died not through natural causes, but from an arsenic-laced communion wafer given to him by a Catholic priest. He believes that Jacques Viogué, a missionary working in Stockholm, administered the poison because he feared Descartes's radical theological ideas would derail an expected conversion to Roman Catholicism by the monarch of Protestant Lutheran Sweden.*Wik

After his death in Stockholm, his body was returned to Paris, arriving on 25 Jun 1665 , though the coffin had been looted by his followers for relics in Stockholm.  Supposedly, the coffin was shipped overland from Copenhagen to avoid piracy by English admirers!  The remains were in Ste. Geneviève, then in Lenoir's Museum of French Monuments, and then finally moved to St‑Germain-des-Prés in 1819. His headstone (or gravestone) is in St‑Germain‑des‑Prés, in the second chapel on the right of the apse.   Stephen Jay Gould says the (purported) skull of Descartes is in the Musée de l'Homme, apparently on display.  Arjen Dijksman recently advised me that the Musee de l'Homme is closed for another year, and there have been efforts to move the skull to the Pantheon.
Église St-Germain-des-Prés, at 3 Place St-Germain-des-Prés, is the oldest church in Paris. Part of it dates to the 6th century, when a Benedictine abbey was founded on the site by King Childebert, son of Clovis. The church was originally built to house a relic of the True Cross brought from Spain in 542. The Normans destroyed the abbey on multiple occasions and only the marble columns in the triforium remain from the original structure. The carved capitals on the pillars are copies of the originals, which are kept in the Musée National du Moyen-Age. The church was enlarged and reconsecrated by Pope Alexander III in 1163. The abbey was completely destroyed during the Revolution, but the church was spared. The present building is a fine example of Romanesque architecture, with gothic interior elements. The square tower dating from the early 11th century, is topped by a landmark spire, which dates to the 19th century. For a time, the abbey served as a pantheon for Merovingian kings. The Chapelle Saint Symphorien, built during the Middle Ages and restored in 1981, served as the necropolis mérovingienne (crypt of the Merovingians). This is the presumed site of first tomb of Saint Germain, Bishop of Paris, who died in 576. Among the others interred here are King Jean-Casimir of Poland

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1712 Brook Taylor suggested that if two glass plates which are clamped together into a “V” are placed into a pan of water then capillary action will draw water up into the shape of a rectangular hyperbola with asymptotes the surface of the water and the point of the “V.” This and several similar experiments performed by Francis Hauksbee before the Royal Society caused Newton to rethink his ideas on capillary force. *VFR

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1730 Euler observes in a letter to Goldbach that 10⁴ + 1 is divisible by 37, and that
3⁸ +2 ⁸ is divisible by 17. Euler cannot prove that any number is the sum of four squares. He has found another result by Fermat, namely that 1 is the only triangular number that is a fourth power (Several years earlier, Goldbach had sent an erroneous proof of this claim to D. Bernoulli) *Lemmemeyer, EULER, GOLDBACH, AND “FERMAT’S THEOREM"

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1776 Captain Cook sails from Deptford on his third voyage, in the 'Resolution' with the 'Discovery' *Nat. Maritime Museum ‏@NMMGreenwich

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1783  Antonie Lavoisier announced to the French Academy of Sciences that water was the product formed by the combination of hydrogen and oxygen. However, this discovery had been made earlier by the English chemist Henry Cavendish *TIS

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1795 The Bureau des Longitudes is a French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardization of time-keeping, geodesy and astronomical observation. During the 19th century, it was responsible for synchronizing clocks across the world. It was headed during this time by François Arago and Henri Poincaré. The Bureau now functions as an academy and still meets monthly to discuss topics related to astronomy.
The Bureau was founded by the National Convention after it heard a report drawn up jointly by the Committee of Navy, the Committee of Finances and the Committee of State education. Henri Grégoire had brought to the attention of the National Convention France's failing maritime power and the naval mastery of England, proposing that improvements in navigation would lay the foundations for a renaissance in naval strength. As a result, the Bureau was established with authority over the Paris Observatory and all other astronomical establishments throughout France. The Bureau was charged with taking control of the seas away from the English and improving accuracy when tracking the longitudes of ships through astronomical observations and reliable clocks.
The ten original members of its founding board were:
Geometers:
Joseph-Louis Lagrange;
Pierre-Simon Laplace;
Astronomers:
Joseph Jérôme Lefrançais de Lalande;
Pierre Méchain;
Jean Baptiste Joseph Delambre;
Dominique, comte de Cassini;
Jean-Charles de Borda, who carried out work related to the mechanics of fluids and precursor of Carnot because of his insights on thermodynamics;
Jean-Nicolas Buache, geographer;
Louis Antoine de Bougainville, celebrated navigator; and
Noël Simon Caroché, manufacturer of telescopes.
*Wik

 Poincaré & Arago

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1903 Marie Skłodowska Curie defended her doctoral thesis on radioactive substances at Université de la Sorbonne in Paris on 25 June 1903 – becoming the first woman in France to receive a doctoral degree.

The examination committee expressed the opinion that Curie's findings, including the determination of radium’s atomic weight, represented the greatest scientific contribution ever made in a doctoral thesis.

In the committee’s three members were two future Nobel Prize laureates: Gabriel Lippmann (Physics 1908) and Henri Moissan (Chemistry 1906).

*The Noble Prize

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1973 Last total solar eclipse with a maximum duration of totality longer than 7 minutes between year 0 and 4000 was June 30, 1973. The eclipse was visible in Africa. The next total solar eclipse with a duration of totality longer than 7 minutes will be on 25 June 2150 in the Pacific Ocean. Thereafter it will be 5 July 2168 in the Indian Ocean. Ref. More Mathematical AstronomicalMorsels by Jean Meeus; Willmann-Bell, 2002. *NSEC

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BIRTHS

1864 Walther Hermann Nernst (25 June 1864 – 18 November 1941) German who was one of the founders of modern physical chemistry. In 1889, he devised his theory of electric potential and conduction of electrolytic solutions (the Nernst Equation) and introduced the solubility product to explain precipitation reactions. In 1906, Nernst showed that it is possible to determine the equilibrium constant for a chemical reaction from thermal data, and in so doing he formulated what he himself called the third law of thermodynamics. This states that the entropy, (a thermodynamic measure of disorder in a system), approaches zero as the temperature goes towards absolute zero. For this, he was awarded the 1920 Nobel Prize in Chemistry. In 1918, he explained the H₂-Cl₂ explosion on exposure to light as an atom chain reaction. *TIS

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1905 Rupert Wildt (/ˈvɪlt/; June 25, 1905 – January 9, 1976) was a German-American astronomer.
He was born in Munich, Germany, and grew up in that country during World War I and its aftermath. In 1927 he was awarded a Ph.D. from the University of Berlin. He joined the University of Göttingen, specializing in the properties of atmospheres.
In 1932 he studied the spectra of Jupiter, and other outer planets, and identified certain absorption bands as belonging to the hydrogen-rich compounds of methane and ammonia. The composition appeared consistent with a composition similar to the sun and other stars.
Assuming that the atmosphere was composed of these gases, during the 1940s and 1950s he constructed a model of the structure of these planets. He believed the core of the planets is solid and composed of a mixture of rock and metal, covered by a thick outer shell of ice, overlaid by a dense atmosphere. His model is still widely accepted.
In 1934 he emigrated to the United States, and became a research assistant at Princeton University from 1937 until 1942. He then became an assistant professor at the University of Virginia until 1947, before joining the faculty of the Yale University.
In 1939 he demonstrated that the major source of optical opacity in the Sun's atmosphere is the H- ion, and thus the main source of visible light for the Sun and stars.
From 1965 until 1968 he was president of the Association of Universities for Research in Astronomy. In the period 1966-1968 he also held the post of the chairman of the department of astronomy at Yale, and from 1973 until his death he was professor emeritus. He died in Orleans, Massachusetts.
His awards include the Eddington Medal in 1966. The Asteroid 1953 Rupertwildt is named after him and the crater Wildt on the Moon is also. *Wik

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 1907 Johannes Hans Daniel Jensen (25 June 1907 – 11 February 1973) was a German physicist who proposed the shell theory of nuclear structure of nucleons - protons and neutrons - grouped in onion-like layers of concentric shells. He suggested that the nucleons spun on their own axis while they moved in an orbit within their shell and that certain patterns in the number of nucleons per shell made the nucleus more stable. Scientists already knew that the electrons orbiting the nucleus were arranged in different shells. For his model of the nucleus, Jensen shared the 1963 Nobel Prize in physics (with Maria Goeppert-Mayer, who arrived at the same hypothesis independently in the U.S.; and Eugene P. Wigner for unrelated work.) Through the 1950s, Jensen worked on radioactivity. *TiS

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1928 Alexei Alexeyevich Abrikosov (June 25, 1928 – March 29, 2017) is a Soviet and Russian theoretical physicist whose main contributions are in the field of condensed matter physics. He was awarded the Nobel Prize in Physics in 2003.
In two works in 1952 and 1957, Abrikosov explained how magnetic flux can penetrate a class of superconductors. This class of materials is known as type-II superconductors. The accompanying arrangement of magnetic flux lines is called the Abrikosov vortex lattice.
Abrikosov was awarded the Lenin Prize in 1966, the Fritz London Memorial Prize in 1972, and the USSR State Prize in 1982. In 1989 he received the Landau Prize from the Academy of Sciences, Russia. Two years later, in 1991, Abrikosov was awarded the Sony Corporation’s John Bardeen Award. The same year he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. He is also a member of the Royal Academy of London, a fellow of the American Physical Society, and in 2000 was elected to the prestigious National Academy of Sciences. He was the co-recipient of the 2003 Nobel Prize in Physics, with Vitaly Ginzburg and Anthony James Leggett, for theories about how matter can behave at extremely low temperatures. *Wik

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DEATHS

1671 Giovanni Riccioli (17 April 1598 – 25 June 1671) Italian astronomer who was the first to observe (1650) a double star (two stars so close together that they appear to be one) -  Mizar in Ursa Major, the middle star in the handle of the Big Dipper. He also discovered satellite shadows on Jupiter. In 1651, he assigned the majority of the lunar feature names in current use. He named the more prominent features after famous astronomers, scientists and philosophers, while the large dark and smooth areas he called "seas" or "maria". The lunar seas were named after moods (Seas of Tranquillity, Serenity) or terrestrial phenomena (Sea of Rains, Ocean or Storms) His map was published in Almagestum Novum in1651.*TIS
Riccioli studied seventy-seven objections to the Copernican thesis and after studying them Riccioli said that the weight of argument favored a “geo-heliocentric” hypothesis such as that advocated by the great Danish astronomer Tycho Brahe. Riccioli's preference for Tycho's model illustrates something important about how science is done. While today anti-Copernicans are often portrayed as Einstein characterized them (opposed to rational thinking, opposed to science), Riccioli, perhaps the most prominent of the anti-Copernicans, examined the available evidence diligently and rationally. The conclusion he reached was indeed wrong, but wrong because at that time neither the diffraction of light and the Airy disk, nor the details of the Coriolis effect, were understood. Riccioli's anti-Copernican arguments were so solid that they would become subjects of further investigation in physics, long after the Copernican theory had triumphed over the Tychonic theory.*Christopher M. Graney, Teaching Galileo, Physics Teacher V50,1

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1879 Sir William Fothergill Cooke (4 May 1806 – 25 June 1879) English inventor who worked with Charles Wheatstone in developing electric telegraphy. Of the pair, Cooke contributed a superior business ability, whereas Wheatstone is generally considered the more important of the two in the history of the telegraph. After Cooke attended a demonstration of the use of wire in transmitting messages, he began his own experiments with telegraphy (1836) and formed a partnership with Wheatstone. Their first patent (1837) was impractical because of cost. They demonstrated their five-needle telegraph on 24 July 1837 when they ran a telegraph line along the railway track from Euston to Camden Town able to transmit and successfully receive a message. In 1845, they patented a single-needle electric telegraph. *TIS

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1941 Alfred Pringsheim (2 September 1850 – 25 June 1941). His work in Fourier series, analytic function theory, and continued fractions was a model of the Weierstrassian approach, although he was not a student of Weier­strass. *VFR
In mathematical analysis, Pringsheim studied real and complex functions, following the power-series-approach of the Weierstrass school. Pringsheim published numerous works on the subject of complex analysis, with a focus on the summability theory of infinite series and the boundary behavior of analytic functions.
Pringsheim's theorem concerns the convergence of a power series with non-negative real coefficients. However, Pringsheim's original proof had a flaw (related to uniform convergence), and a correct proof was provided by Ralph P. Boas. Pringsheim's theorem is used in analytic combinatorics and the Perron–Frobenius theory of positive operators on ordered vector spaces.
Besides his research in analysis, Pringsheim also wrote articles for the Encyclopedia of Mathematical Sciences on the fundamentals of arithmetic and on number theory. He published papers in the Annals of Mathematics. As an officer of the Bavarian Academy of Sciences, he recorded the minutes of its scientific meetings.
Pringsheim and Ivan Śleszyński, working separately, proved what is now called the Śleszyński–Pringsheim theorem on convergence of certain continued fractions.*Wik

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 1948 Bento de Jesus Caraça, GCSE, GOL (18 April 1901 – 25 June 1948) was an influential Portuguese mathematician, economist and statistician. Caraça was also a member of the Portuguese Communist Party, and participated in the formation of the Portuguese Movement of National Antifascist Unity and Movement of Democratic Unity in the 1940s.

Caraça was one of the founders of the Portuguese Mathematical Society in 1940, and from 1945–1945 served as joint president alongside Aureliano de Mira Fernandes  Caraça founded the journal Gazeta de Matemática  in 1940 with mathematicians António Aniceto Monteiro, Hugo Ribeiro, José da Silva Paulo and Manuel Zaluar Nunes.

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1960 Walter Baade (24 Mar 1893; 25 Jun 1960 at age 67) German-American astronomer who, with Fritz Zwicky, proposed that supernovae could produce cosmic rays and neutron stars (1934), and Baade made extensive studies of the Crab Nebula and its central star. During WW II blackouts of the Los Angeles area Baade used the 100-inch Hooker telescope to resolve stars in the central region of the Andromeda Galaxy for the first time. This led to his definition of two stellar populations, to the realization that there were two kinds of Cepheid variable stars, and from there to a doubling of the assumed scale of the universe. Baade and Rudolph Minkowski identified and took spectrograms of optical counterparts of many of the first-discovered radio sources, including Cygnus A and Cassiopeia A. *TIS

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1974 Cornelius Lanczos (2 Feb 1893 - 25 June 1974) worked on relativity and mathematical physics and invented what is now called the Fast Fourier Transform. *SAU Lanczos served as assistant to Albert Einstein during the period of 1928–29.*Wik

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1978 Hsien Chung Wang (April 18, 1918 — June 25, 1978.)worked on algebraic topology and discovered the 'Wang sequence', an exact sequence involving homology groups associated with fibre bundles over spheres. These discoveries were made while he worked with Newman in Manchester. Wang also solved, at that time, an important open problem in determining the closed subgroups of maximal rank in a compact Lie group *SAU

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  Ernest Thomas Sinton Walton (6 October 1903 – 25 June 1995) was an Irish physicist and Nobel laureate who first split the atom. He is best known for his work with John Cockcroft to construct one of the earliest types of particle accelerator, the Cockcroft–Walton generator. In experiments performed at Cambridge University in the early 1930s using the generator, Walton and Cockcroft became the first team to use a particle beam to transform one element to another. According to their Nobel Prize citation: "Thus, for the first time, a nuclear transmutation was produced by means entirely under human control". *Wik

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1997 Jacques-Yves Cousteau (11 June 1910 – 25 June 1997) French naval officer, oceanographer, marine biologist and ocean explorer, known for his extensive underseas investigations. He was co-inventor of the aqualung which made SCUBA diving possible (1943). Cousteau developed the Conshelf series of manned habitats, the Diving Saucer, a process of underwater television and numerous other platforms and specialized instruments of ocean science. In 1945 he founded the French Navy's Undersea Research Group. He modified a WWII wooden hull minesweeper into the research vessel Calypso, in 1950. An observation dome added to the foot of Calypso's bow was found to increase the ship's stability, speed and fuel efficiency. *TIS

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2006 Irving "Kap" Kaplansky (March 22, 1917, Toronto – June 25, 2006, Los Angeles) was born in Toronto, Ontario, Canada after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D  from Harvard in 1941 as Saunders Mac Lane's first student, Kaplansky was professor of mathematics at the University of Chicago from 1945 to 1984. He was chair of the department from 1962 to 1967.
"Kap," as his friends and colleagues called him, made major contributions to group theory, ring theory, the theory of operator algebras and field theory. He published over 150 papers with over 20 co-authors. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He was the Director of the Mathematical Sciences Research Institute from 1984 to 1992, and the President of the American Mathematical Society from 1985 to 1986.
Kaplansky also was a noted pianist known to take part in Chicago performances of Gilbert and Sullivan productions. He often composed music based on mathematical themes. One of those compositions, A Song About Pi, is a melody based on assigning notes to the first 14 decimal places of pi.
Kaplansky was the father of singer-songwriter Lucy Kaplansky, who occasionally performs A Song About Pi in her act.
He was among the first five recipients of William Lowell Putnam fellowships in 1938.*Wik

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2014  Lonnie Grafton Cross (May 22, 1927 – June 25, 2014) was an African American mathematician who took the name Abdulalim Abdullah Shabazz. With an outstanding record working to increase the participation of women, minorities, and individuals with physical disabilities into science and engineering, he was awarded the Mentor Award of the American Association for the Advancement of Science and the Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring. *SAU

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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