Tuesday 31 October 2023

Soul Cakes, Halloween Began in Britain??

Well, you get old and eventually you learn stuff. I got a note from Charles Wells who said, " The *name* Halloween came from the British. It is the eve of All Saints Day (Hallowmas, November 1) which is celebrated all over Catholic Europe, not just in Britain. November 2 is All Souls Day, meaning the day for sinners as well as saints. That is the Day of the Dead in Mexico."
Later I added, "Thanks, Charles,but given that the Scottish, the English, the Welsh, and the Irish ALL seem to object to being included as "British", I will simply confirm what you have said with a quote from the Online Etymology Dictionary, "c.1745, Scottish shortening of Allhallow-even "Eve of All Saints, last night of October" (1556), the last night of the year in the old Celtic calendar, where it was Old Year's Night, a night for witches. Another pagan holiday given a cursory baptism and sent on its way. Hallowmas "All-saints" is first attested 1389."

 
I had just seen a BBC show that morning in 2009 and Sting had just released the Soul Cakes song, don't ask why, some things must be left unexplained.
 

 Here is the story of soul cakes and halloween as told by Wikipedia: "A Soul cake is a small round cake which is traditionally made for All Souls' Day to celebrate the dead. The cakes, often simply referred to as souls, were given out to soulers (mainly consisting of children and the poor) who would go from door to door on Hallowmas (new word to me, obviously the eve of All Souls Day) singing and saying prayers for the dead. Each cake eaten would represent a soul being freed from Purgatory. The practice of giving and eating soul cakes is often seen as the origin of modern Trick or Treating." "The tradition of giving Soul Cakes originated in Britain during the Middle Ages, although similar practices for the souls of the dead were found as far south as Italy." "The cakes were usually filled with allspice, nutmeg, cinnamon, or other sweet spices, raisins or currants, and later were topped with the mark of a cross. They were traditionally set out with glasses of wine on All Hallows Eve, and on All Saints Day children would go "souling" by calling out: Soul, Soul, a soul cake! 
 I pray thee, good missus, a soul cake! 
 One for Peter, two for Paul, 
 three for Him what made us all! 
 Soul Cake, soul cake, please good missus, a soul cake.
 An apple, a pear, a plum, or a cherry,
 anything good thing to make us all merry. 
 One for Peter, one for Paul, 
& three for Him who made us all. ...

lyrics from A Soalin', a holiday song written and performed by Peter, Paul and Mary (1963)."

See, nothing scary here. Have a Happy Halloween.

On This Day in Math - October 31

  


*The image is a pumpkin carved a few Halloweens ago by Sonja L. One of my Stats/Calc students (and a really good Bassoonist, Bassooner, Bassoon-enough)

I don’t like reciprocals. 
They’re always trying to get one over on you.

The 304th day of the year; 304 is the sum of six consecutive primes starting with 41, and also the sum of eight consecutive primes starting with 23. (and for those who keep up with such things, it is also the record number of wickets taken in an English cricket season by Tich Freeman in 1928.)

There are 304 semi-primes less than 2^10, but 304 is NOT one of them. *Derek Orr 

304 = 77^2 - 75^2 = 23^2 - 15^2 = 40^2 - 36^2

Math Joke for Halloween: Why do mathematicians confuse Halloween and Christmas? Because Oct 31 = Dec 25 (31 in base 8 (Octal) is the same quantity as 25 in Decimal)



EVENTS

In 1815, English chemist, Sir Humphrey Davy of London (Davy was actually from Penzance) patented the miner's safety lamp. Miners at work constantly met firedamp, an explosive mix of methane gas and air, during the working of coal. This was an almost insurmountable obstacle to the working of many of the collieries until the discovery of the safety lamp. The flame of the safety lamp is surrounded by a copper or iron gauze cylinder, with openings no more than 1/24-inch. Such a fine gauze prevents flame passing through, but fails if coarser. The wire absorbs or conducts away the heat of the flame contained inside the lamp so it does not explode gas outside the lamp. If firedamp is present, a pale blue flame appears around the central flame. This warns a miner to leave the area immediately! *TIS
Davy Statue at Penzance




1839 (Sometime in October) the first teacher’s institute was held at Hartford, Connecticut, 26 men teachers attended a six week course sponsored by Henry Barnard and received the “opportunity of critically reviewing the studies which they will be called upon to teach, with a full explanation of all the principles involved.” The authority who gave instruction on higher mathematics was Charles Davies. *VFR
Charles Davies (January 22, 1798 – September 17, 1876) was a professor of mathematics at the United States Military Academy, notable for writing a series of mathematical textbooks.




1903 At a New York meeting of the AMS F. N. Cole (1861-1927) presented a paper “On the factoring of large numbers.” He spoke not a word, but carefully raised 2 to the 67th power, then subtracted one. Moving over he computed 193,707,721 times 761,838,257,287. The calculations agreed, showing that 2^67 − 1 was not a Mersenne prime. E. T. Bell, in Mathematics—Queen and Servant of the Sciences, wrote, with his usual exageration, “For the first and only time on record, an audience of the American Mathematical Society vigorously applauded the author or a paper delivered before it.” Later, in 1911, Bell asked Cole how long it had taken him to find this factorization and he replied “Three years of Sundays.” It is instructive to check this arithmetic on your hand held calculator. [Eves, Adieu, 297◦; BAMS 10(1903), 134] *VFR   
 Édouard Lucas had demonstrated in 1876 that M67 must have factors (i.e., is not prime), but he was unable to determine what those factors were


1915 Closing date for a prize consisting of a gold medal bearing the portrait of Weierstrass and 3000 Swedish crowns for the best essay on the theory of analytic functions. King Gustav V of Sweden founded the prize to commemorate the centenary of the birth of Weierstrass. *VFR

1918 The wife of the Russian mathematician Lyapunov died of tuberculosis. On the same day, Lyapunov shot himself. He died three days later, on 3 November 1918. *VFR

1931   Herman Hupfeld wrote a well known song for the Broadway musical Everybody's Welcome which opened on October 31, 1931. In the original show, it was sung by Frances Williams. It was first recorded by Rudy Vallée on July 25, 1931 for Victor Records, then also by Jacques Renard and his Orchestra on Brunswick Records and Fred Rich. It was at first a modest hit, but was voted the second best song on the AFI's best songs of film.   You know it, but if I give you the opening lyrics, you may have never heard them.  But when I get to the chorus, you'll know.  
This day and age we're living in 
Gives cause for apprehension, 
With speed and new invention 
And things like fourth dimension. 
Yet we get a trifle weary 
With mr. einstein's theory, 
So we must get down to earth at times: 
Relax, relieve the tension. 
And no matter what the progress 
Or what may yet be proved, 
The simple facts of life are such 
They cannot be removed. 

You must remember this, 
A kiss is still a kiss,
.........As Time Goes By 
------------------------------------------------------------------------------------------------------------

1933 Albert Einstein moved to the United States on 17 October 1933. Two weeks later, Halloween arrived. When a group of girls knocked on his door that evening and shouted 'Trick or Treat,' Einstein came to the front porch and played the violin for them: * @phalpern

In 1992, the Vatican admitted erring for over 359 years in formally condemning Galileo Galilei for entertaining scientific truths such as the Earth revolves around the sun it, which the Roman Catholic Church long denounced as anti-scriptural heresy. After 13 years of inquiry, the Pope's commission of historic, scientific and theological scholars brought the pope a "not guilty" finding for Galileo. *TIS In 1822 the church lifted the ban on the works of Galileo and in 1979 Pope John Paul II selected a commission to investigate. On Mar 31 of 1984 the Vatican newspaper, L’Observatore Romano, stated, “The so-called heresy of Galileo does not seem to have any foundation, neither theologically nor under canon law.” It still took until Oct 31, 1992, before Pope John Paul II declared that the church may have been mistaken in condemning Galileo. *Wik

(I can see the Papal Bull now, "Oops, Our Bad!")






BIRTHS

1711 Laura Maria Catarina Bassi (31 Oct 1711 in Bologna, Papal States, 20 Feb 1778 in Bologna, Papal States) was an Italian physicist and one of the earliest women to gain a position in an Italian university. *SAU She was the first woman in the world to earn a university chair in a scientific field of studies. She received a doctoral degree from the University of Bologna in May 1732, only the third academic qualification ever bestowed on a woman by a European university, and the first woman to earn a professorship in physics at a university in Europe. She was the first woman to be offered an official teaching position at a university in Europe.
In 1738, she married Giuseppe Veratti, a fellow academic with whom she had twelve children. After this, she was able to lecture from home on a regular basis and successfully petitioned the University for more responsibility and a higher salary to allow her to purchase her own equipment.
One of her principal patrons was Pope Benedict XIV. He supported less censorship of scholarly work, such as happened with Galileo, and he supported women figures in learning, including Agnesi.
She was mainly interested in Newtonian physics and taught courses on the subject for 28 years. She was one of the key figures in introducing Newton's ideas of physics and natural philosophy to Italy. She also carried out experiments of her own in all aspects of physics. In order to teach Newtonian physics and Franklinian electricity, topics that were not focused in the university curriculum, Bassi gave private lessons.[6] In her lifetime, she authored 28 papers, the vast majority of these on physics and hydraulics, though she did not write any books. She published only four of her papers.[2] Although only a limited number of her scientific works were left behind, much of her scientific impact is evident through her many correspondents including Voltaire, Francesco Algarotti, Roger Boscovich, Charles Bonnet, Jean Antoine Nollet, Giambattista Beccaria, Paolo Frisi, Alessandro Volta. Voltaire once wrote to her saying "There is no Bassi in London, and I would be much happier to be added to your Academy of Bologna than that of the English, even though it has produced a Newton". *Wik



1815 Karl (Theodor Wilhelm) Weierstrass (31 Oct 1815; 19 Feb 1897) was a German mathematician who is known as the "father of modern analysis" for his rigor in analysis led to the modern theory of functions, and considered one of the greatest mathematics teachers of all-time. He was doing mathematical research while a secondary school teacher, when in 1854, he published a paper on Abelian functions in the famous Crelle Journal. The paper so impressed the mathematical community that he shortly received an honorary doctorate and by 1856, he had a University appointment in Berlin. In 1871, he demonstrated that there exist continuous functions in an interval which have no derivatives nowhere in the interval. He also did outstanding work on complex variables. *TIS
*Wik



1847 Galileo Ferraris (31 Oct 1847; 7 Feb 1897) Italian physicist who studied optics, acoustics and several fields of electrotechnics, but his most important discovery was the rotating magnetic field. He produced the field with two electromagnets in perpendicular planes, and each supplied with a current that was 90º out of phase. This could induce a current in a incorporated copper rotor, producing a motor powered by alternating current. He produced his first induction motor (with 4 poles) in May-Jun 1885. Its principles are now applied in the majority of today's a.c. motors, yet he refused to patent his invention, and preferred to place it at the service of everyone. *TIS

1890 Joseph Jean Camille Pérès (31 Oct 1890 in Clermont-Ferrand, France, 12 Feb 1962 in Paris, France) Pérès' work on analysis and mechanics was always influenced by Volterra, extending results of Volterra's on integral equations. His work in this area is now of relatively little importance since perhaps even for its day it was somewhat old fashioned.
A joint collaboration between Pérès and Volterra led to the first volume of Theorie generale des fonctionnelles published in 1936. Although the project was intended to lead to further volumes only this one was ever published. This work is discussed in where the author points out that the book belongs to an older tradition, being based on ideas introduced by Volterra himself from 1887 onwards. By the time the work was published the ideas it contained were no longer in the mainstream of development of functional analysis since topological and algebraic concepts introduced by Banach, von Neumann, Stone and others were determining the direction of the subject. However, the analysis which Pérès and Volterra studied proved important in developing ideas of mathematical physics rather than analysis and Pérès made good use of them in his applications. *SAU

1902 Abraham Wald (October 31, 1902 – December 13, 1950) was a mathematician born in Cluj, in the then Austria–Hungary (present-day Romania) who contributed to decision theory, geometry, and econometrics, and founded the field of statistical sequential analysis. He spent his researching years at Columbia University.*Wik

1919 Father Magnus J. Wenninger OSB (born Park Falls, Wisconsin, October 31, 1919) is a mathematician who works on constructing polyhedron models, and wrote the first book on their construction. *Wik

1925 John A. Pople (31 Oct 1925; 15 Mar 2004) British mathematician and chemist who, (with Walter Kohn), received the 1998 Nobel Prize in Chemistry for his work on computational methodology to study the quantum mechanics of molecules, their properties and how they act together in chemical reactions. Using Schrödinger's fundamental laws of quantum mechanics, he developed a computer program which, when provided with particulars of a molecule or a chemical reaction, outputs a description of the properties of that molecule or how a chemical reaction may take place - often used to illustrate or explain the results of different kinds of experiment. Pople provided his GAUSSIAN computer program to researchers (first published in 1970). Further developed, it is now used by thousands of chemists the world over. *TIS

1927 Narinder Singh Kapany (31 Oct 1927, )Indian-American physicist who is widely acknowledged as the father of fibre optics. He coined the term fibre optics for the technology transmitting light through fine glass strands in devices from endoscopy to high-capacity telephone lines that has changed the medical, communications and business worlds. While growing up in Dehradun in northern India, a teacher informed him that light only traveled in a straight line. He took this as a challenge and made the study of light his life work, initially at Imperial College, London. On 2 Jan 1954, Nature published his report of successfully transmitting images through fiber optical bundles. The following year he went to the U.S. to teach. In 1960, Optics Technology. He holds over 100 patents.*TIS

1935 Ronald Lewis Graham (born October 31, 1935- Jul 6 2020) is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. Graham was also featured in Ripley's Believe It or Not for being not only "one of the world's foremost mathematicians", but also "a highly skilled trampolinist and juggler", and past president of the International Jugglers' Association. He is currently the Chief Scientist at the California Institute for Telecommunications and Information Technology (also known as Cal-(IT)2) and the Irwin and Joan Jacobs Professor in Computer Science and Engineering at the University of California, San Diego. *Wik My current favorite Graham quote is, "An ideal math talk should contain one proof and one joke and they should not be the same."

Graham died of bronchiectasis on July 6, 2020, at the age of 84.

Ron, his wife Fan Chung, and his friend, Paul Erdos





DEATHS

*@willgater
1867 William Parsons, 3rd Earl of Rosse (17 Jun 1800, 31 Oct 1867) was an Irish astronomer who built the largest reflecting telescope of the 19th century. He learned to polish metal mirrors (1827) and spent the next few years building a 36-inch telescope. He later completed a giant 72-inch telescope (1845) which he named "Leviathan," It remained the largest ever built until decades after his death. He was the first to resolve the spiral shape of objects - previously seen as only clouds - which were much later identified as galaxies independent of our own Milky Way galaxy and millions of light-years away. His first such sighting was made in 1845, and by 1850 he had discovered 13 more. In 1848, he found and named the Crab Nebula (because he thought it resembled a crab), by which name it is still known. *TIS A reproduction of the Leviathan of Parsonstown (now Birr) Ireland at Birr Castle, Rosse’s ancestral home and is open to tourists.

1899 Juliusz Paweł Schauder (September 21, 1899, Lwów, Austria-Hungary – October, 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics.
He had to fight in World War I right after his graduation from school. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów.
Schauder was Jewish, and after the invasion of German troops in Lwów it was impossible for him to continue his work. In his letters to Swiss mathematicians, he wrote that he had important new results, but no paper to write them down. He was executed by the Gestapo, probably in October 1943.
Most of his mathematical work belongs to the field of functional analysis, being part of a large Polish group of mathematicians, i.e. Lwów School of Mathematics. They were pioneers in this area with wide applications in all parts of modern analysis. Schauder is best known for the Schauder fixed point theorem which is a major tool to prove the existence of solutions in various problems, the Schauder bases (a generalization of an orthonormal basis from Hilbert spaces to Banach spaces), and the Leray−Schauder principle, a way to establish solutions of partial differential equations from a priori estimates. *Wik

1988 George Eugene Uhlenbeck (6 Dec 1900, 31 Oct 1988) Dutch-American physicist who, with Samuel A. Goudsmit, proposed the concept of electron spin (Jan 1925) - a fourth quantum number which was a half integer. This provided Wolfgang Pauli's anticipated "fourth quantum number." In their experiment, a horizontal beam of silver atoms travelling through a vertical magnetic field was deflected in two directions according to the interaction of their spin (either "up" or "down") with the magnetic field. This was the first demonstration of this quantum effect, and an early confirmation of quantum theory. As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure, the kinetic theory of matter and extended Boltzmann's equation to dense gases.*TIS


Credits
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*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 30 October 2023

Notes on the History of Graph Paper

Direct repost from 2011 to preserve notes.If you have corrections, comments, or additions, please send me a note, comment, telegram.....



 



Just re-ordered graph paper for next year for my department. We don't use nearly as much these days as five or ten years ago... calculators have made them much less common in schools. It reminded me that I hadn't actually put anything here about my notes on the history of graph paper, so for those who are interested...

Graph paper, a math class staple, was developed between 1890 and 1910. During this period the number of high school students in the U.S. quadrupled, and by 1920, according to E.L Thorndike, one of every three teenagers in America “enters High School”, compared to one in ten in 1890. The population of “high school age” people had also grown so that the total number of people entering HS was six times as great as only three decades before. Research mathematicians and educators took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.” When the University of Chicago opened in 1892 E.H. Moore was the acting head of the mathematics department. “Moore was born in Marietta, Ohio, in 1862, and graduated from Woodward High School in Cincinnati. “(from Milestones in (Ohio) Mathematics, by David E. Kullman) Moore was President of the American Mathematical Society in 1902. The Fourth Yearbook of the NCTM, Significant Changes and Trends in the Teaching of Mathematics Throughout the World Since 1910, published in 1929, has on page 159, “The graph, of great and growing importance, began to receive the attention of mathematics teachers during the first decade of the present century (20th)” . Later on page 160 they continue, “The graph appeared somewhat prior to 108, and although used to excess for a time, has held its position about as long and as successfully as any proposed reform. Owing to the prominence of the statistical graph, and the increased interest in educational statistics, graphic work is assured a permanent place in our courses in mathematics.” [emphasis added]
Hall and Stevens “A school Arithmetic”, printed in 1919, has a chapter on graphing on “squared paper”.

Using google's n-gram viewer I arrived at the conclusion that from 1880 until appx 1925 the term square paper was the most popular with coordinate paper close behind.  The earliest mention of graph paper in relation to math education was in Advanced Algebra by Joseph Victor Collins.  I found a use in 1890 of logarithmic graph paper (defined as "Ordinate scales printed on logarithmic graph paper" in a report " Geological Survey Water-supply Paper - Issues 1890-1894").  The term may have been more used in engineering fields before it was adapted in mathematics.  However the shift occurred, by 1930 "Graph paper" was the most common term, and by 1940 it was more common than the other two terms combined, and by 1960 it reached eight times the usage of either of the others.  *PB 

John Bibby has written (August,2012) to advise me that John Perry, who was at the time President of the Institution of Electrical Engineers, has a section on "Use of Squared Paper" in an article in Nature in 1900 (The teaching of mathematics, Nature Aug 1900 pp.317-320.) They wanted $19 to see the article, so I take John at his word. I did find another similar endorsement of "squared paper" by Perry in "Englands Neglect of Science" published in 1900 also. On page 18 after several lamentations about trained engineers who had no ability/understanding of the mathematics applying to their field, he writes: "I tell you, gentlemen, that there is only one remedy for this sort of thing. Just as the antiquated method of studying arithmetic has been given up, so the antiquated method of studying other parts of mathematics must be given up. The practical engineer needs to use squared paper." 

The actual first commercially published “coordinate paper” is usually attributed to Dr. Buxton of England in 1795 (if you know more about this man, let me know). The earliest record I know of the use of coordinate paper in published research was in 1800. Luke Howard (who is remembered for creating the names of clouds.. cumulus, nimbus, and such) included a graph of barometric variations. [On a periodical variation of the barometer, apparently due to the influence of the sun and moon on the atmosphere. Philosophical Magazine, 7 :355-363. ]
[The above was gathered from a numbur of authoritive sources including a Smithsonian site, but on a recent visit to Monticello, the home of my longtime favorite American Prisident, Thomas Jefferson, I discovered it was in error. I found a use by Jefferson in his use of the paper for architectural drawings earlier than any of these dates. Here is the information from the Moticello web site.]
Prior to 1784, when Jefferson arrived in France, most if not all of his drawings were made in ink. In Paris, Jefferson began to use pencil for drawing, and adopted the use of coordinate, or graph, paper. He treasured the coordinate paper that he brought back to the United States with him and used it sparingly over the course of many years. He gave a few sheets to his good friend David Rittenhouse, the astronomer and inventor:

"I send for your acceptance some sheets of drawing-paper, which being laid off in squares representing feet or what you please, saves the necessity of using the rule and dividers in all rectangular draughts and those whose angles have their sines and cosines in the proportion of any integral numbers. Using a black lead pencil the lines are very visible, and easily effaced with Indian rubber to be used for any other draught." {Jefferson to David Rittenhouse, March 19, 1791}
A few precious sheets of the paper survive today.

 

The increased use of graphs and graph paper around the turn of the century is supported by a Preface to the “New Edition” of Algebra for Beginners by Hall and Knight. The book, which was reprinted yearly between the original edition and 1904 had no graphs appearing anywhere. When the “New Edition” appeared in 1906 it had an appendix on “Easy Graphs”, and the cover had been changed to include the subhead, “Including Easy Graphs”. The preface includes a strong statement that “the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shews that anything on a smaller scale (such as ‘millimeter’ paper) is practically worthless in the hands of beginners.” He finishes with the admonition that, “The growing fashion of introducing graphs into all kinds of elementary work, where they are not wanted, and where they serve no purpose – either in illustration of guiding principles or in curtailing calculation – cannot be too strongly deprecated. (H. S. Hall, 1906)” The appendix continued to be the only place where graphs appeared as late as the 1928 edition. The term “graph paper seems not to have caught on quickly. I have a Hall (the same H S Hall as before) and Stevens, A school Arithmetic, printed in 1919 that has a chapter on graphing on “squared paper”. Even later is a 1937 D. C. Heath text, Analytic Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase “coordinate paper” (page 223, topic 153). Even in 1919 Practical mathematics for Home Study by Claude Irwin Palmer introduced a section on “Area Found by the Use of Squared Paper” and then defined “paper accurately ruled into small squares” (pg 183). It may be that the term squared paper hung on much longer in England than in the US. I have a 1961 copy of Public School Arithmetic (“Thirty-sixth impression, First published in 1910) by Baker and Bourne published in London that still uses the term “squared paper” but uses graphs extensively.

I recently found one other earlier use of coordinate grid on paper. The Metropolitan Museum of Art owns a pattern book dated to around 1596 in which each page bears a grid printed with a woodblock. The owner has used these grids to create block pictures in black and white and in color.


Of course "graph paper" could not have preceded the term "graph" for a curve of a function relationship, and many teachers and students might be surprised to know that it was not until 1886 when George Chrystal wrote in his Algebra I, "This curve we may call the graph of the function." The actual first known use of the term "graph" for a mathematical object actually predates this event by only eight years and occurred in a discrete math topic.   J. J. Sylvester published a note in February 1878 using 'graph' to denote a set of points connected by lines to represent chemical connections. In that note "Chemistry and Algebra", Sylvester
wrote: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph" .


A more or less famous Kekule structure is the benzine shown at right.
(August Kekule von Stradonitz was one of the founders of structural organic chemistry, and is remembered for his dreams of the structure of benzene as a snake swallowing its own tail.)

This short note in Nature was more a notice of the more complete paper he had written in American Journal of Mathematics, Pure and Applied, would appeared the same month,  "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices,"  The term "graph" first appears in this paper on page 65.  The images he uses appear below. 





Graph has come to have multiple meanings in mathematics, but for most students it relates to the graph of functions on the coordinate axes.  The origin is from the Greek graphon, to write, perhaps with earlier references to carving or scratching. Jeff Miller's web site suggests that the use of graph as a verb may have first been introduced as late as 1898. 
In a post to a history newsgroup, Karen Dee Michalowicz commented on the history of graphing:
It is interesting to note that the coordinate geometry that Descartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later.  In fact, I find it appearing in the mid 1800's in my old college texts in analytic geometry.  It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. [This matches rise of  graph paper in the same periods].  Graphing is most often found in books by Wentworth.  Even so, the texts written in the 20th century, perhaps until the 1960's, did not all have graphing.  Taking Algebra I in the middle 1950's, I did not learn to graph until I took Algebra II

Math historian Bea Lumpkin has written about the early graphs by the Egyptians in what was an early use of what painters call the grid method:
In my article ... I suggest, "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square.  It needs just one short, important step from the use of square grids to the location of points by coordinates.  
In the same posting she comments on the finding of graphs in Egyptian finds dating back to 2700 BC: 
"An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara.  It is a limestone flake, apparently complete, measuring about 5 x 7 x 2 inches, inscribed on one face in red ink, and probably belongs to the III rd dynasty"  Here is the reason that Clark and Engelbach attached great importance to the diagram.  It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally.  The vertical coordinates are given in cubits, palms and fingers.  The horizontal spacing, the authors write "... most probably that is to be understood as one cubit, an implied unit elsewhere."  To clinch their analysis, Clarke and Engelbach observe:  "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated tot he curve obtained from the data on the ostrakon. 


This certainly lays claim to the oldest line graph I have ever heard.  

On This Day in Math - October 30

  


'Mathematics is the science that uses easy words for hard ideas.'
~ Edward Kasner

The 303rd day of the year; there are 303 different bipartite graphs with 8 vertices. *What's Special About This Number

303 primes are below 2000. * Derek Orr

303 = 152^2 - 151^2 = 52^2 - 49^2

In the Gregorian calendar, 303 is the number of years that are not leap years in a period of 400 years.

EVENTS

1613 Kepler married his second wife (the first died of typhus). She was fifth on his slate of eleven candidates. The story that he used astrology in the choice is doubtful.*VFR Kepler married the 24-year-old Susanna Reuttinger. He wrote that she, "won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren.According to Kepler's biographers, this was a much happier marriage than his first. *Wik
The complete story from a letter of Kepler about his 2 year search is told here by Thony Christie.

1710 William Whiston, whom Newton had arranged to succeeded him as Lucasian Professor at Cambridge in 1701, was deprived of the chair and driven from Cambridge for his unorthodox religious views. Whiston was removed from his position at Cambridge, and denied membership in the Royal Society for his “heretical” views. He took the “wrong” side in the battle between Arianism (a unitarian view) and the Trinitarian view, but his brilliance still made the public attend to his proclamations. When he predicted the end of the world by a collision with a comet in October 16th of 1736 the Archbishop of Canterbury had to issue a denial to calm the panic (VFR put it this way, "it is not acceptable to be a unitarian at the College of the Whole and Undivided Trinity".)
His translation of the works of Flavius Josephus may have contained a version of the famous Josephus Problem, and in 1702 Whiston's Euclid discusses the classic problem of the Rope Round the Earth, (if one foot of additional length is added, how high will the rope be). I am not sure of the dimensions in Whiston's problem, and would welcome input, I have searched the book and can not find the problem in it, but David Singmaster has said it is there, and he is not an easy source to reject. It is said that Ludwig Wittgenstein was fascinated by the problem and used to pose it to students regularly.





1735 Benjamin Franklin’s paper “On the Usefulness of Mathematics,” appeared in the Pennsylvania Gazette. [NCTM yearbook # 32(1970), p. 20]*VFR I have also seen the date given as October 30. Some historians also question whether or not this was actually written by Franklin.

1826 Abel presented a paper to the French Academy of Science that was ignored by Cauchy, who was to serve as referee. The paper was published some twenty years later.*VFR

In 1937, the closest approach to the earth by an asteroid, Hermes, was measured to be 485,000 miles, which, to an astronomer, is a mere hair's width (asteroid now lost).*TIS
On Oct. 28, 1937, astronomer Karl Reinmuth of Heidelberg noticed an odd streak of light in a picture he had just taken of the night sky. About as bright as a 9th magnitude star, it was an asteroid, close to Earth and moving fast--so fast that he named it Hermes, the herald of Olympian gods. On Oct. 30, 1937, Hermes glided past Earth only twice as far away as the Moon, racing across the sky at a rate of 5 degrees per hour. Nowadays only meteors and Earth-orbiting satellites move faster.

Plenty of asteroids were known in 1937, but most were plodding members of the asteroid belt far beyond Mars. Hermes was different. It visited the inner solar system. It crossed Earth's orbit. It proved that asteroids could come perilously close to our planet. And when they came, they came fast.
Reinmuth observed Hermes for five days. Then, to make a long story short, he lost it.

(don't worry, they found it!)
mid October, 2003:  The asteroid Hermes was re-discovered last week after being lost for
66 years. Now Jean-Luc Margot, a researcher in the Department of
Earth and Space Sciences at the University of California, Los
Angeles, has determined that the asteroid is in fact two objects
orbiting each other. The two objects together would cover an area
approximately the size of Disneyland.
Hermes approaches Earth's orbit twice every 777 days. Usually our planet is far away when the orbit crossing happens, but in 1937, 1942, 1954, 1974 and 1986, Hermes came harrowingly close to Earth itself. We know about most of these encounters only because Lowell Observatory astronomer Brian Skiff re-discovered Hermes… on Oct. 15, 2003. Astronomers around the world have been tracking it carefully ever since. Orbit-specialists Steve Chesley and Paul Chodas of NASA's Jet Propulsion Laboratory (JPL) have used the new observations to trace Hermes' path backwards in time, and so they identified all the unnoticed flybys.




1945 The first conference on Digital Computer Technique was held at MIT. The conference was sponsored by the National Research Council, Subcommittee Z on Calculating Machines and Computation. Attended by the Whirlwind team,(The Whirlwind computer was developed at the Massachusetts Institute of Technology. It is the first computer that operated in real time, used video displays for output, and the first that was not simply an electronic replacement of older mechanical systems) it influenced the direction of this computer. *CHM

1978 Laura Nickel and Curt Noll, eighteen year old students at California State at Hayward, show that 221,701 − 1 is prime. This was the largest prime known at that time. *VFR (By Feb of the next year, Noll had found another, 223209-1. By April, another larger Prime had been found.)

1992 The Vatican announced that a 13-year investigation into the Catholic Church’s condemnation of Galileo in 1633 will come to an end and that Galileo was right: The Copernican Theory, in which the Earth moves around the Sun, is correct and they erred in condemning Galileo. *New York Times for 31 October 1992.

2012 After Hurricane Sandy came ashore in New Jersey on the 29th, the huge weather system was captured with an overlay to emphasize it's Fibonacci-like structure. *HT to Bob Mrotek for sending me this image




BIRTHS
1840 Joseph Jean Baptiste Neuberg (30 Oct 1840 in Luxembourg City, Luxembourg - 22 March 1926 in Liège, Belgium) Neuberg worked on the geometry of the triangle, discovering many interesting new details but no large new theory. Pelseneer writes, "The considerable body of his work is scattered among a large number of articles for journals; in it the influence of A Möbius is clear." *SAU
In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is named after Neuberg , who first introduced the curve in a paper published in 1884.The curve appears as the first item, with identification number K001, in Bernard Gilbert's Catalogue of Triangle Cubics which is a compilation of extensive information about more than 1200 triangle cubics.
There are several ways to determine the Locus of points in the curve, but Neuberg used determinants,  






1844 George Henri Halphen (30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He did his studies at École Polytechnique (X 1862). He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry.*Wik

1863 Stanislaw Zaremba (3 Oct 1863 in Romanowka, Poland - 23 Nov 1942 in Kraków, Poland) From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. Zaremba played a crucial role in this transformation. Much of Zaremba's research work was in partial differential equations and potential theory. He also made major contributions to mathematical physics and to crystallography. He made important contributions to the study of viscoelastic materials around 1905. He showed how to make tensorial definitions of stress rate that were invariant to spin and thus were suitable for use in relations between the stress history and the deformation history of a material. He studied elliptic equations and in particular contributed to the Dirichlet principle.*SAU

1906 Andrei Nikolaevich Tikhonov (30 Oct 1906 in Gzhatska, Smolensk, Russia - November 8, 1993, Moscow) Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935. These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics.
The extremely deep investigations of Tikhonov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.
Tikhonov's work on mathematical physics continued throughout the 1940s and he was awarded the State Prize for this work in 1953. However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative. After a series of fundamental papers introducing the topic, the work was carried on by his students.
Another area in which Tikhonov made fundamental contributions was that of computational mathematics. Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ... and other branches of the natural sciences were evolved and put into practice. ... One of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes, which Tikhonov developed in collaboration with Samarskii.
In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems. He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems. Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin Prize for his work on ill-posed problems in 1966. In the same year he was elected to full membership of the USSR Academy of Sciences.*SAU

1907 Harold Davenport (30 Oct 1907 in Huncoat, Lancashire, England - 9 June 1969 in Cambridge, Cambridgeshire, England) Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He wrote a number of important textbooks and monographs including The higher arithmetic (1952)*SAU

1946 William Paul Thurston  (October 30, 1946 – August 21, 2012) American mathematician who was awarded the Fields Medal in 1983 for his work in topology. As early as his Ph.D. thesis entitled Foliations of 3-manifolds which are circle bundles (1972) that showed the existence of compact leaves in foliations of 3-manifolds, Thurston had been working in the field of topology. In the following years, Thurston's contributions to the field of foliations were recognized to be of considerable depth, set apart by their originality. This was also true of his subsequent work on Teichmüller space. *TIS



DEATHS

1626 Willebrord Snel van Royen l (13 June 1580 in Leiden, Netherlands - 30 Oct 1626 in Leiden, Netherlands) Snell was a Dutch mathematician who is best known for the law of refraction, a basis of modern geometric optics; but this only become known after his death when Huygens published it. His father was Rudolph Snell (1546-1613), the professor of mathematics at Leiden. Snell also improved the classical method of calculating approximate values of π by polygons which he published in Cyclometricus (1621). Using his method 96 sided polygons gives π correct to 7 places while the classical method yields only 2 places. Van Ceulen's 35 places could be found with polygons of 230 sides rather than 262. In fact Van Ceulen's 35 places of π appear in print for the first time in this book by Snell. *SAU





1631 Michael Mästin (30 Sept 1550 in Göppingen, Baden-Würtemberg, Germany
- 30 Oct 1631 in Tübingen, Baden-Würtemberg, Germany) astronomer who was Kepler's teacher and who publicized the Copernican system. Michael Mästin was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centered orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU

1739 Leonty Filippovich Magnitsky (June 9, 1669, Ostashkov – October 30, 1739, Moscow) was a Russian mathematician and educator. From 1701 and until his death, he taught arithmetic, geometry and trigonometry at the Moscow School of Mathematics and Navigation, becoming its director in 1716. In 1703, Magnitsky wrote his famous Arithmetic (Арифметика; 2,400 copies), which was used as the principal textbook on mathematics in Russia until the middle of the 18th century. This book was more an encyclopedia of mathematics than a textbook because most of its content was communicated for the first time in Russian literature. In 1703, Magnitsky also produced a Russian edition of Adriaan Vlacq's log tables called Таблицы логарифмов и синусов, тангенсов и секансов (Tables of Logarithms, Sines, Tangents, and Secants). Legend has it that Leonty Magnitsky was nicknamed Magnitsky by Peter the Great, who considered him a "people's magnet" *Wik

1805 Ormbsy MacKnight Mitchel (July 20, 1805 – October 30, 1862) American astronomer and major general in the American Civil War.
A multi-talented man, he was also an attorney, surveyor, and publisher. He is notable for publishing the first magazine in the United States devoted to astronomy. Known in the Union Army as "Old Stars", he is best known for ordering the raid that became famous as the Great Locomotive Chase during the Civil War. He was a classmate of Robert E. Lee and Joseph E. Johnston at West Point where he stayed as assistant professor of mathematics for three years after graduation.
The U.S. communities of Mitchell, Indiana, Mitchelville, South Carolina, and Fort Mitchell, Kentucky were named for him. A persistently bright region near the Mars south pole that was first observed by Mitchel in 1846 is also named in his honor. *TIA

1806 Alexander (Dallas) Bache (July 19, 1806 – February 17, 1867) was Ben Franklin's great grandson. A West Point trained physicist, Bache became the second Superintendent of the Coast Survey (1844-65). He made an ingenious estimate of ocean depth in 1856. He studied records of a tidal wave that had taken 12 hours to cross the Pacific. Knowing that wave speeds depend on depth, he calculated a 2 1/5-mile average depth for the Pacific (within 15% of the right value). Bache created the National Academy of Sciences, securing greater government involvement in science. Through the Franklin Institute he instituted boiler tests to promote safety for steamboats.*TIS

1975 Gustav Hertz (22 July 1887, 30 Oct 1975) German quantum physicist who, with James Franck, received the Nobel Prize for Physics in 1925 for the Franck-Hertz experiment, which confirmed the quantum theory that energy can be absorbed by an atom only in definite amounts and provided an important confirmation of the Bohr atomic model. He was a nephew of Heinrich Hertz. Although he fought on the German side in World War I, being of Jewish descent, he was forced to resign his professorship (1934) when Hitler took power. From 1945 he worked in the Soviet Union, and then in 1955 was a professor of physics in Leipzig, East Germany.*TIS




2007 Juha Heinonen, (23 July 1960 in Toivakka, Finland - 30 Oct 2007 in Ann Arbor, Michigan, USA) Professor of Mathematics passed away on October 30. He arrived in the Department in 1988 as a postdoctoral assistant professor, and became a professor in 2000. He was a leading researcher in geometric function theory, having published two books and numerous articles with many collaborators. Most recently, Juha served as Associate Chair for Graduate Studies in the Department, where he mentored many young mathematicians. *Math at U of M webpage memorial (Heinonen died at the age of 47 'after a brief but courageous battle with kidney cancer'. The Department of Mathematics at the University of Michigan established the Juha Heinonen Memorial Graduate Student Fellowship in his honour. An international conference in his memory Quasiconformal Mappings and Analysis on Metric Spaces was organised at the University of Michigan, Ann Arbor in May 2008.)


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