"The great mathematician Euclid is said to have told his students 'There is no royal road to geometry." Thus begins an otherwise nice article in the Canadian Globe and Mail newspaper on line by Anna Stokke. The article describes her concern with what she sees as a failure of the "hands-on, manipulative approach" to math education in Alberta, and across Canada. After thirty years of teaching, I share many of her concerns. Read the article for yourself here.

If Ms Stokke was just any journalist, I might commend her for her (somewhat sketchy) math history connection. But she is NOT just any journalist. She is an assistant professor at the University of Winnipeg, and a co-founder of the non-profit organization Archimedes Math School. (I know absolutely nothing about the Archimedes Math School except that it is named for an ancient Greek Mathematician and therefore suggests a connection to math-historical knowledge.)

Perhaps the requirements of producing text for print required editing the first quote down to a triviality, and so I can not be too critical of a bit of historical vagueness without knowing the nature of her task better than I do. For most of her readers, I am sure the omission went without note and provided a little verbal quip to support the idea of greater analytic rigor in their children's education.

I, on the other hand, am retired, write mostly for teachers and students, and fully believe that one of the things that build the interest in mathematical studies for students are stories that make the math, and the mathematicians come alive. Just as a million kids grew to learn and love baseball sitting on the couch with dad or mom watching the home team, hearing their stories of the heroes of their youth, and maybe even memories of their own exploits. They lean the history, and the culture of the practice, and so are more willing to spend time learning to do it well. Few kids complain about Dad helping them learn to hit better. To me, math teachers have a mini-opportunity to do the same thing by mixing stories of the history and development of mathematics, the false starts, and the great insights, and the mysterious connections that intertwine mathematical (and scientific and social) topics.

So if you are a student learning the culture, or a teacher who wants to share it, here is a little bit more that might be told about the cast off "quip" at the start of Ms Stokke's column:

So, in the manor of Ms. Stokke, I will begin with a well known quote, ""Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds." Yeah you know that one; the U. S. Postal Service motto that is inscribed on the James Farley Post Office in New York City. You can see part of it above the beautiful Corinthian Columns in this photo from Wikipedia.

Farley was the 53rd Postmaster General of the United States, but he didn't create the quote. He may not have even ever have heard of it. It was supplied by one of the architects who designed the building, and then carved into the face by a designer/artist who would go on to become a designer at DC Comics. (do you think we have the kids attention yet?)

The quote itself was from a history written by Herodotus, a 5th century BC Greek. The couriers he was speaking of were not the US Postal Service, but the riders on Persian King Darius I road throughout his empire. Herodotus added, " "There is nothing in the world that travels faster than these Persian couriers." This "Royal Road" throughout the Persian Empire was over 1600 miles long, and the riders could cover it in 7 days; a very early Pony Express.

Let one hundred plus years pass and Alexander the Great has a general named Ptolemy who decides when Alexander dies to make himself the ruler of Egypt, Ptolemy I. Also in Alexandria about this time was a mathematician who was putting together all the mathematical knowledge of the Greeks into a set of "Elements" which could be used to derive other mathematical knowledge. Ptolemy was a big fan, but a busy man, and he found the Elements difficult to digest.

AND..... It was this student, Ptolemy I, whose continued requests for an "easier" way to learn the Elements" that supposedly moved Euclid to remind him that there was no "Royal Road", such as the one still stretching across Persia at that time, to speed the learning of mathematics.

Did it ever really happen? Maybe not. The first known record of the event comes over five centuries later at the pen of Proclus, around 450 AD. Even if he never said it, we imagine he would have.

I usually closed this story by reminding my students of the lost guy driving in New York City looking for Carnegie Hall as the hour of his concert approached. The seemingly empty streets held little hope when he saw a vagrant looking fellow leaning against the wall of a building, eyes closed. He tapped his horn and when the guy opened his eyes, asked, " Can you tell me how to get to Carnegie Hall?"

The vagrant shook his head a moment, eyes closed, then opened them again to declare , "You gotta' practice man, you gotta' really practice."

## Thursday 31 October 2013

### Not Just ANY Student, and not just any Royal Road

Labels:
Darius,
Euclid,
royal road

## Wednesday 30 October 2013

### Great Problems for High School

Sometimes I come across problems that make me wish I was teaching High School again. I mean I don't want to grade papers or go to staff meetings or get up every morning at 5am like I used to; but the idea of watching a bright class of kids thinking about a problem that is just different enough to make them use some of the skills they have other than their great memory was always exciting, and I guess I'll never get over it.

Ever once in awhile I come across a problem that makes me want to pop them on a HS class, but since I'm retired and too lazy to go back to work full time, I thought I would share a couple of the recent ones with the folks still out there working the front lines in case you may have missed them with all the papers to grade and such.

Not too long ago James Tanton reminded me of an old problem about chess boards by giving a new one I had never seen. The classic problem I refer to is the one that asks, "Is it possible to cover an 8x8 chess board with dominoes (1x2 rectangles) if the opposite corner squares are cut off. You, and probably your clever students have already seen this, but it is still a clever problem because of the symmetry idea in the solution. Tanton threw out the question, "In tiling an 8x8 board with dominos, must there be two dominos making a 2x2 square?"

Now at this point I admit I haven't even solved the question, and I'm not sure if Professor Tanton has, (but bet he has). My first instinct is that there must, but I haven't spent the time to test the "Why?" of that. You see, that would spoil it a little. I guess I eventually will, but presenting it to a class when you DON'T know the answer makes the discovery even more exciting. You let the students work without the temptation to rush in and "guide them" to an answer. My experiences in such situations always made me proud of the kinds of thinking my kids could produce when the problem was not "textbook".

Another I saw recently was on Greg Ross' Futility Closet blog. He posted "(5/8)

^{2}+ 3/8 = (3/8)

^{2}+ 5/8."

Now giving this to students is not actually a question, unless they have a mathematical mind, in which case they will ask the question; "What does this imply?" For me the immediate question looked like two fractions a/c and b/c so that if you added either to the square of the other the results would be equal. Now the numbers a,b, and c that make that happen would be the question of interest. This might be at a slightly different level than the previous problem, but I keep thinking both would be appropriate from 7th grade to the last year of High School.

The most recent is from John Allen Paulos twitter where he posted a complete proof in the 140 characters allowed. The question was prove that there must be some irrational numbers a and b so that a

^{b}is rational. I'll give you his solution to this one because it is one of a couple that should be exposed to advanced level math students in HS so that they can see some of the beautiful proofs that are out there:

Paulos proof: expanded beyond 140 characters for greater readability, Let a and b both be sqrt2 (irrat.)

Now it may be that c=a

^{b}is rational. If it is, we are done; but if not, then c

^{ b }= 2.

That is not one that will pop out as easy to most students. I played with it and decided that I would try to explain it like this:

sqrt(2) = 2

^{1/2}so sqrt(2)^sqrt(2) = 2

^{1/2sqrt(2)}and using the product of powers we can get 2

^{sqrt(2)/2}. By using sqrt(2)/2 = 1/sqrt(2) we finally arrive at sqrt(2)^sqrt(2)= 2

^{1/sqrt(2)}.

I'm thinking that after this they will be able to see that raising that to the power of sqrt(2) will give a result of 2, a perfectly rational number.

I think at this stage in their lives many of them find rational, irrational, transcendental, imaginary and such a bit mystifying, and some controlled experiments let them gain a little confidence. I'm reminded of a recent blog I read where a teacher/researcher talking to two kids sitting across from each other asked one if she knew a name for the shape in front of her. It was a triangle arraigned so that from her view it was a nabla (∇)(had she been at all aware of the word). She replied that she only knew that from where here classmate was sitting across the table, it would be a triangle. Imagine how many times she must have seen a triangle without seeing one in different orientations. True learning spins on such delicate wheels. You can say the terms as many times as you wish, but when you get your students to describe their views of them, they may start to get a deeper understanding of what they are dealing with.

And if time allows, it is always a wonderfully amazing thing to walk students through the remarkable fact that i

^{i}is a real, and infact, is equal to e

^{-π/2}. Do point out that this is real, but not rational.

One disappointment with both of these examples is they really illustrate that there exists a number a of a certain type (irrational for instance) such that a

^{a}has a certain property (rational for instance) . I think it would be nice to have a collection of irrational pairs a,b such that a

^{b}are rational and demonstrably so at a high school level. Would love to hear your suggestions?

One final one just because it demonstrates how beautifully geometry can make some math problems visual. This one is also from Greg Ross.

The question is, "How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?" The question reminds me of Kirkman's schoolgirl problem which involved 15 girls walking in groups.

The geometric solution is easy if you start with the committees as lines in a plane so that no three are concurrent. Then each of the six intersections represents a person and the problem is solved.

OK, Just one more. A short time ago I came across a neat trick by Martin Gardner in Ivars Peterson's column, the Mathematical Tourist that I had never seen, and I thought I had read Gardner's stuff.

And since all kids love Mobius strips, and this one was even new and surprising to me, I thought I would share.

Start with a simple cross of paper (make it kind of large as some cutting is involved) as shown in the illustration at right.

Take one cross and do the usual half-twist to make a Mobius Strip. The other is just made into a conventional loop with no twist.

NOW, trisect the Mobius band, and bisect the normal loop.. shake it all out... and be amazed. Then share it with kids..

Labels:
chess board,
fraction curio,
paulos,
symmetry,
twitter proof

### On This Day in Math - October 30

**'Mathematics is the science that uses easy words for hard ideas.'**

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

**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

**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**Ben Franklin published “On the Usefulness of Mathematics,” his only published article on mathematics. *VFR

**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

**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 2

^{21,701}− 1 is prime. This was the largest prime known at that time. *VFR (By Feb of the next year, Noll had found another, 2

^{23209}-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

**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 van Royen Snell**(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

## Tuesday 29 October 2013

### The Abacus and Counting Frame in American Education, a Brief History

The idea that all children should learn arithmetic seems to have blossomed in the western countries around 1800. Prior to this period arithmetic training had been reserved for a small group of boys bound for mechanical or business fields. The training didn't usually begin until at least the age of twelve, and featured rote memorization of the written numerals and the rules for special cases to solve arithmetic operations.

Michalowicz and Howard (2003) have argued that “students of the 18th century rarely had a textbook,” that those who studied arithmetic “wrote in a ‘cipher’ book,” and that “the textbook was mainly for the teacher or for individuals who were self-taught.”pg 79).

Another writer quoted an educator in Boston around 1810 as stating that “printed arithmetics were not used in the Boston schools” until after he left there (p. 45). Rather, teachers set “sums” for

their pupils out of ciphering books that they had prepared at school, or had copied from textbooks or from the ciphering books of other teachers.(Monroe, 1912, pp. 5-16).

When mass arithmetic education started to become popular in the British Ilse and US, these rote memorization approaches continued and were used on even younger students. One of the first educators to influence Britain, and the US away from this structured approach toward a "mental/experiential" approach to understanding arithmetic was the famous Swiss educator, J. H. Pestalozzi. One of the tools he used as a primary instructional item was the horizontal abacus, or counting frame. But the path that brought it from its vertical Roman roots to the horizontal classroom model had a long and winding route that balanced on a narrow turn of events in the life of a French mathematician/soldier in the Napoleonic campaign in Russia. And his is the story I wish to tell here.

The abacus has been around, in one form or another, since at least the ancient Greeks and Romans. The image at the top appears on a second-century CE funerary relief that depicts the deceased young man reclining beneath his dead father's portrait, with his grieving mother seated on the right. The slave standing on the left is operating an abacus, which symbolizes the family's success in business. (Barbara F. McManus, The Roman Abacus

*on the web*)

It spread throughout the European and Asian continents and by 1300 was common throughout both continents. Then the sweeping adaptation of Arabic numerals and improved computational methods led to its complete disappearance in western Europe, so that by the late 18th century it was unremembered. But in the later part of the 18th century (my best guess) a horizontal form of the Roman abacus became common for Russian classrooms (who apparently discovered the idea of "understanding arithmetic" slightly earlier). These abaci, called the schoty (счёты) were not only horizontal, the wire frames bowed out forward of the frame allowing the teacher to hold them up in front of a class without disturbing the order. In addition they were made with the fifth and sixth beads colored differently to make it easier to recognize numbers.

The single row of four beads was for working with a fractional quarter-ruble coin that existed at that time. Why it was place four rows from the bottom (commonly) has never been explained to me.

So the stage is set for an 1809 graduate of the E'cole Polytechnique in Paris, and student of Gaspard Monge to return to his home in Metz in the Alsace-Lorraine region. Then in 1812, he was called to duty with Napoleon's forces to invade Russia. If you don't remember, that didn't go well for Napoleon, and not too well for our young mathematician either. He was captured in November of that year, and sent off on a forced march of "hundreds of miles". Keep in mind that Russia was cold and snowy that November, and weary soldiers on forced marches were prone to, and in fact did die.... but not our hearty hero. For over a year, he was kept in a Russian prison, and while there decided to reconstruct and improve and some old ideas of his teacher Monge, and the great Lazare Carnot. These writings would become a classic work in projective geometry, Traite' des propiertes projectives des figures (1822). When he was freed and repatriated to France, he returned to his home in Metz, and brought along a Russian abacus. He gave it to a teacher in Metz, and suggested that it might be useful in teaching small children. The item had been so forgotten that it was treated as a novelty as it slowly began to be reintroduced in France, then more quickly into Britain and the US under supporters of Pestalozzi.

And the weary warrior/mathematician whose survival made it all possible? If you didn't get the clue with the title to his classic work in projective geometry, it was Jean Victor Poncelet. As a mathematician, his most notable work was in projective geometry, in particular, his work on Feuerbach's theorem. He also made discoveries about projective harmonic conjugates; among these were the poles and polar lines associated with conic sections. These discoveries led to the principle of duality, and also aided in the development of complex numbers and projective geometry. And if you ever happen to be visiting the Eiffel Tower, look up, there are 72 names of scientists around the 1st stage of the tower, and yes, our hero/warrior/mathematician is one of them.

If you teach, tell this story to your students. His greatest influence may not be the mathematical writing he was recognized for, but a small item that he passed along to a teacher in his home town thinking, "It might be useful for teaching children."

John Golden points out in a comment that a small type of counting frame is becomming very popular in elementary education again. It is called the rekenrek (which seems to have been semi-americanized from the Dutch rekentuig for abacus. The root is the same Germanic root that gives us the English term "reckon" for counting or doing arithmetic. Interestingly, its proto-Germanic root was for "motion in a straight line" which seems perfect for a counting frame.

The version I have seen promoted has only two horizontal wires, but there seem to be some available with multiple rows as well.

### On This Day in Math - October 29

**Allez en avant, et la foi vous viendra**

**Push on and faith will catch up with you.**

The 302nd day of the year; There are 302 ways to play the first three moves in checkers.

**EVENTS**

**1669**Newton, aged twenty-six, appointed Lucasian Professor at Cambridge. This post required Newton to lecture once each week on “some part of Geometry, Astronomy, Geography, Optics, Statics, or some other Mathematical discipline,” and to deposit ten of those lectures in the library each year. The students were required to attend, but like all other requirements they ignored this one too. We know of only three people who attended a lecture at Cambridge by Newton. [Westfall 208–210; Works, 3, xv] *VFR

**1675**Leibniz ﬁrst used the integral sign. Also ﬁrst used “d”. He also constructed what he calls the “triangulum characteristicum,” which had been used before him by Pascal and Barrow. [Cajori, History of Mathematical Notations, vol. 2, p. 2; Struik’s Source Book mistakenly has 26 October]

VFR Historical notes for the calculus classroom ,

In these same pages he will write examples of the integrals of x

^{2}and x

^{3},and then illustrate that a constant multiple may be taken outside the integral as shown in the image below.

On the left is Liebniz integral sign with a vincula in place of todays parentheses to show that he is integrating the quantity (a/b) l Then the open bottomed box is Liebniz symbol for equality,then he shows the constant (a/b) multiplied by the integral of l .

Although he was not the first to use the idea, Jeff Miller's web site on the first use of math words has, "The term CHARACTERISTIC TRIANGLE was used by Leibniz and apparently coined by him, as triangulum characteristicum."

**1878**Patent issued for Odhner calculating machine. *VFR Willigot T. Odhner was granted a patent for a calculating machine that performed multiplications by repeated additions. The patent, a modified and compact version of Gottfried von Leibniz stepped wheel, was acquired and embodied in Brunsviga calculators that sold into 1950s.*CHM

**1929**"Black Tuesday", the great USA stock market crash. About 16 million shares were traded, and the Dow lost an additional 30 points, or 12%.. "Anyone who bought stocks in mid-1929 and held onto them saw most of his or her adult life pass by before getting back to even." Richard M. Salsman *Wik

1964 Asteroid "Lucifer" is discovered by astronomer Elizabeth Roemer. amhistorymuseum @amhistorymuseum Roemer was the winner of the 1946 Science Talent Search and is now Professor Emerita, Lunar and Planetary Laboratory, University of Arizona. *Smithsonian Institution Archives

1985 On October 29th, 1985, the 329th birthday of Edmond Halley, the British threw a big party in honor of the return of Halley's Comet. The Halley's Comet Royal Gala was held at Wembley Conference Centre, London. It was a combination Variety Show and "Who's Who" in British Society, hosted by Princess Anne of the British Royal Family. *Joseph M. Laufer, Halley's Comet Society, USA

**In 1991**, space probe Galileo become the first human object to fly past an asteroid, Gaspra, making its closest approach at a distance of 1,604 km, passing at a speed of 8 km/sec (5 mi/sec). The encounter provided much data, including 150 images, which showed Gaspra has numerous craters indicating it has suffered numerous collisions since its formation. Gaspra is about 20-km long and orbits the Sun in the main asteroid belt between Mars and Jupiter. Gaspra, asteroid 951, was discovered by Ukrainian astronomer Grigoriy N. Neujamin (1916) who named it after a Black Sea retreat. In the photograph (left), subtle color variations have been exaggerated by NASA to highlight changes in reflectivity, surface structure and composition. *TIS

1998, Nearly four decades after he became the first American to orbit Earth, John Glenn is relaunched into space. *@HISTORYmag

**BIRTHS**

Pitman described himself as 'a mathematician who strayed into Statistics'; nevertheless, his contributions to statistical and probability theory were substantial.

Pitman was active in the formation of the Australian Mathematical Society in 1956. He also took an active part in the Summer Research Institutes organized by the Mathematical Society, and used them as a sounding board for his research on statistical inference.

He was a renowned member of the Statistical Society of Australia, attending its biennial conferences. In 1978 the Statistical society established the Pitman Medal.

Pitman presented the first systematic account of non-parametric inference and lectured extensively on the subject, both in Australia and in the United States. The kernel of the subject, as described by him, is 'Suppose that the sum of two samples A, B is the sample C. Then A, B are discordant if A is an unlikely sample from C.' Again, he writes, 'The approach to the subject, starting from the sample and working towards the population instead of the reverse, may be a bit of a novelty'; and later, 'the essential point of the method is that we do not have to worry about the populations which we do not know, but only about the sample values which we do know'.

The notes of the 'Lectures on Non-parametric Inference' given in the United States, though never published, have been widely circulated and have had a major impact on the development of the subject. Among the new concepts introduced in these Lectures are asymptotic power, efficacy, and asymptotic relative efficiency.

A major contribution to probability theory is his elegant treatment of the behavior of the characteristic function in the neighborhood of the origin, in three papers. This governs such properties as the existence of moments. There are also interesting properties of the Cauchy distribution, and of subexponential distributions.

On his death, on 21 July 1993, Edwin was buried at the Hobart Regional Cemetery in Kingston. He lives on in the memory of many of us who are grateful for his life and legacy.

*Evan J. Williams, Australian Academy of Science

**1925 Klaus Friedrich Roth**(29 Oct 1925, )German-born British mathematician who was awarded the Fields Medal in 1958. His major work has been in number theory, particularly the analytic theory of numbers. He solved in the famous Thue-Siegel problem (1955) concerning the approximation to algebraic numbers by rational numbers (for which he won the medal). Roth also proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turán of 1935).*TIS

**DEATHS**

**1783 Jean le Rond D'Alembert**(16 Nov 1717, 29 Oct 1783) was abandoned by his parents on the steps of Saint Jean le Rond, which was the baptistery of Notre-Dame, qv in Section 7-A-1. Foster parents were found and he was christened with the name of the saint. [Eves, vol. II, pp. 32 33. Okey, p. 297.] When he became famous, his mother attempted to reclaim him, but he rejected her. *VFR Known for his work in various fields of applied mathematics, in particular dynamics. In 1743 he published his Traité de dynamique (Treatise on Dynamics). The d'Alembert principle extends Newton's third law of motion, that Newton's law holds not only for fixed bodies but also for free moving bodies. D'Alembert also wrote on fluid dynamics, the theory of winds, the properties of vibrating strings and conducted experiments on the properties of sound . His most significant purely mathematical innovation was his invention and development of the theory of partial differential equations. He published eight volumes of mathematical studies (1761-80). He was editor of the mathematical and scientific articles for Denis Diderot's Encyclopédie.*TIS

**1917 Giovanni Battista Guccia**(21 Oct 1855 in Palermo, Italy - 29 Oct 1914 in Palermo, Italy) Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves. His results published in volume one of the Rendiconti del Circolo Matematico di Palermo were extended by Corrado Segre in 1888 and Castelnuovo in 1897. *SAU

**1921 Konstantin Alekseevich Andreev**(26 March 1848 in Moscow, Russia - 29 Oct 1921 Near Sevastopol, Crimea) Andreev is best known for his work on geometry, although he also made contributions to analysis. In the area of geometry he did major pieces of work on projective geometry. Let us note one particular piece of work for which he has not received the credit he deserves. Gram determinants were introduced by J P Gram in 1879 but Andreev invented them independently in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions. *SAU

**1931 Gabriel Xavier Paul Koenigs**(17 January 1858 Toulouse, France – 29 October 1931 Paris, France) was a French mathematician who worked on analysis and geometry. He was elected as Secretary General of the Executive Committee of the International Mathematical Union after the first world war, and used his position to exclude countries with whom France had been at war from the mathematical congresses.

He was awarded the Poncelet Prize for 1913.*Wik

**1933 Paul Painlevé**worked on differential equations. He served twice as prime-minister of France. *SAU

**1951 Robert Aitken**(31 Dec 1864, 29 Oct 1951) American astronomer who specialized in the study of double stars, of which he discovered more than 3,000. He worked at the Lick Observatory from 1895 to 1935, becoming director from 1930. Aitken made systematic surveys of binary stars, measuring their positions visually. His massive New General Catalogue of Double Stars within 120 degrees of the North Pole allowed orbit determinations which increased astronomers' knowledge of stellar masses. He also measured positions of comets and planetary satellites and computed orbits. He wrote an important book on binary stars, and he lectured and wrote widely for the public. *TIS

**1993 Lipman Bers**(May 22, 1914 – October 29, 1993) was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.*Wik

**1993 Robert Palmer Dilworth**(December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem (Dilworth 1950) relating chains and antichains in partial orders; he was also the first to study antimatroids (Dilworth 1940). Dilworth advised 17 Ph.D. students and as of 2010 has 373 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist.*Wik

Credits :

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*RMAT= The Renaissance Mathematicus, Thony Christie

*SAU=St Andrews Univ. Math History

*TIA = Today in Astronomy

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

## Monday 28 October 2013

### On This Day in Math - October 28

"Big Fleas have little fleas upon their backs to bite 'em,

and little fleas have lesser fleas,

and so ad infinitum.

The 301st day of the year; 301 is the sum of three consecutive primes starting at 97

**EVENTS**

**1386**Opening of the University of Heidelberg. It is the oldest university in Germany and was the third university established in the Holy Roman Empire. *Wik

**1462**Archbishop Adolph of Nassau captured the city of Maintz and allowed his soldiers to plunder the city. This forced Gutenberg and his printers to ﬂee, but rather than nipping printing in the bud, it forced its spread to Strasburg, Cologne, Basel, Augsburg, Ulm, Nuremberg, Subiaco, and by 1470, Paris. [G. H. Putnam, Books and Their Makers During the Middle Ages (1896),

p. 372]. *VFR

**1636**Harvard College founded. The only mathematical master’s thesis in the U.S. before 1700 was at Harvard. This was in 1693 when the candidate took the aﬃrmative position on “Is the quadrature of a circle possible?”. *VFR

**1886**The Statue of Liberty was dedicated on Bedloe’s Island in New York Harbor. The sculptur Bartholin was present. The statue had almost been moved to another city when there was not enough interest in New York to pay the cost of building the pedestal. Joseph Pulitzer, publisher of the World, a New York newspaper, announced a drive to raise $100,000 (the equivalent of $2.3 million today). Pulitzer pledged to print the name of every contributor, no matter how small the amount given.The drive captured the imagination of New Yorkers, especially when Pulitzer began publishing the notes he received from contributors. "A young girl alone in the world" donated "60 cents, the result of self denial." As the donations flooded in, the committee resumed work on the pedestal. After five months of daily calls to donate to the statue fund, on August 11, 1885, the World announced that $102,000 had been raised from 120,000 donors, and that 80 percent of the total had been received in sums of less than one dollar. *Wik

1957 Only three weeks after Sputnik went into space, young Denis Cox in Victoria, Australia sent a design for a spaceship addressed, "TO A TOP SCIENTIST AT Woomera ROCKET RANGE South Australia." His design included locations for Austalian Insignia, four Rolls Royce Engines, guided missiles, etc, but advised the scientists, "YOU PUT IN OTHER DETAILS". The letter can be seen here at the Letters of Note web site Edited by Shaun Usher.

On September 24, 2009, an article on ABC Australia's web page indicated that "The Defence Science Technology Organisation is now finally organising a letter from rocket scientists in response to the letter."

**In 1965**, the Gateway Arch (630' (190m) high) was completed in St. Louis, Missouri. This graceful sweeping tapered curve of stainless steel is the tallest memorial in the U.S. The architect of the catenary curve arch was Eero Saarinen who won the design competition in 1947. It was constructed 1961-66 in the Jefferson National Expansion Memorial Park, established on the banks of the Mississippi River, on 21 Dec 1935, to commemorate the westward growth of the United States between 1803 and 1890. Cost for the $30 million national monument was shared by the federal government and the City of St. Louis. The memorial arch has an observation room at the top for visitors reached by trams running inside the legs of the arch.*TIS

**BIRTHS**

**1703 Antoine Deparcieux**(28 Oct 1703 in Clotet-de-Cessous, France - 2 Sept 1768 in Paris, France) was a French mathematician who is best known for an early work on annuities and mortality.*SAU

**1804 Pierre François Verhulst**(28 October 1804, Brussels, Belgium – 15 February 1849, Brussels, Belgium) was a mathematician and a doctor in number theory from the University of Ghent in 1825. Verhulst published in 1838 the equation:

dN/dt = r N (1-N/k)

when N(t) represents number of individuals at time t, r the intrinsic growth rate and k is the carrying capacity, or the maximum number of individuals that the environment can support. In a paper published in 1845 he called the solution to this the logistic function, and the equation is now called the logistic equation. This model was rediscovered in 1920 by Raymond Pearl and Lowell Reed, who promoted its wide and indiscriminate use.*Wik

**1845 Ulisse Dini**(14 Nov 1845 in Pisa, Italy - 28 Oct 1918 in Pisa, Italy) Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet. He discovered a condition, now known as the Dini condition, ensuring the convergence of a Fourier series in terms of the convergence of a definite integral. As well as trigonometric series, Dini studied results on potential theory. *SAU

**1880 Michele Cipolla**(born 28 October 1880 in Palermo; died 7 September 1947 in Palermo) was an Italian mathematician, mainly specializing in number theory.

He was a professor of Algebraic Analysis at the University of Catania and, later, the University of Palermo. He developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the problem of binomial congruence.*Wik

**1937 Dr. Marcian Edward (Ted) Hoff, Jr**. was born October 28, 1937 at Rochester, New York. He received a BEE (1958) from Rensselear Polytechnic Institute in Troy, NY. During the summers away from college he worked for General Railway Signal Company in Rochester where he made developments that produced his first two patents. He attended Stanford as a National Science Foundation Fellow and received a MS (1959) and Ph.D. (1962) in electrical engineering. He joined Intel in 1962. In 1980, he was named the first Intel Fellow, the highest technical position in the company. He spent a brief time as VP for Technology with Atari in the early 1980s and is currently VP and Chief Technical Officer with Teklicon, Inc. Other honors include the Stuart Ballantine Medal from the Franklin Institute.*CHM

**1955 Bill Gates,**cofounder and CEO of Microsoft Corporation, was born. Gates developed a version of BASIC for the Altair 8800 while being a student at Harvard. With the success of BASIC, he and co-developer Paul Allen founded Microsoft, which delivered an operating system for the IBM PC, the Microsoft Word word processing program, the Window system software, and other programs. *CHM

**DEATHS**

**1703 John Wallis**(23 Nov 1616, 28 Oct 1703) British mathematician who introduced the infinity math symbol. Wallis was skilled in cryptography and decoded Royalist messages for the Parliamentarians during the Civil War. Subsequently, he was appointed to the Savilian Chair of geometry at Oxford in 1649, a position he held until his death more than 50 years later. Wallis was part of a group interested in natural and experimental science which became the Royal Society, so Wallis is a founder member of the Royal Society and one of its first Fellows. Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton. *TIS

**1916 Cleveland Abbe**(3 Dec 1838, 28 Oct 1916) U.S. astronomer and first meteorologist, born in New York City, the "father of the U.S. Weather Bureau," which was later renamed the National Weather Service. Abbe inaugurated a private weather reporting and warning service at Cincinnati. His weather reports or bulletins began to be issued on Sept. 1, 1869. The Weather Service of the United States was authorized by Congress on 9 Feb 1870, and placed under the direction of the Signal Service. Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather.*TIS

1918 Edward Bouchet (15 Sept 1852, New Haven, Conn – 28 Oct 1918, New Haven, Conn) was the first African-American to earn a Ph.D. in Physics from an American university and the first African-American to graduate from Yale University in 1874. He completed his dissertation in Yale's Ph.D. program in 1876 becoming the first African-American to receive a Ph.D. (in any subject). His area of study was Physics. Bouchet was also the first African-American to be elected to Phi Beta Kappa.

Bouchet was also among 20 Americans (of any race) to receive a Ph.D. in physics and was the sixth to earn a Ph. D. in physics from Yale.

When Bouchet was born there were only three schools in New Haven open to black children. Bouchet was enrolled in the Artisan Street Colored School with only one teacher, who nurtured Bouchet's academic abilities. He attended the New Haven High School from 1866–1868 and then Hopkins School from 1868-1870 where he was named valedictorian (after graduating first in his class).

Bouchet was unable to find a university teaching position after college, most likely due racial discrimination. Bouchet moved to Philadelphia in 1876 and took a position at the Institute for Colored Youth (ICY). He taught physics and chemistry at the ICY for 26 years. The ICY was later renamed Cheyney University. He resigned in 1902 at the height of the W. E. B. Du Bois-Booker T. Washington controversy over the need for an industrial vs. collegiate education for blacks.

Bouchet spent the next 14 years holding a variety of jobs around the country. Between 1905 and 1908, Bouchet was director of academics at St. Paul's Normal and Industrial School in Lawrenceville, Virginia (presently, St. Paul's College). He was then principal and teacher at Lincoln High School in Gallipolis, Ohio from 1908 to 1913. He joined the faculty of Bishop College in Marshall, Texas in 1913. Illness finally forced him to retire in 1916 and he moved back to New Haven. He died there, in his childhood home, in 1918, at age of 66. He had never married and had no children.*Wik

**1924 John Backus**(3 Dec 1924, 28 Oct 1988) American computer scientist who invented the FORTRAN (FORmula TRANslation) programming language in the mid 1950s. He had previously developed an assembly language for IBM's 701 computer when he suggested the development of a compiler and higher level language for the IBM 704. As the first high-level computer programming language, FORTRAN was able to convert standard mathematical formulas and expressions into the binary code used by computers. Thus a non-specialist could write a program in familiar words and symbols, and different computers could use programs generated in the same language. This paved the way for other computer languages such as COBOL, ALGOL and BASIC. *TIS

**1965 Luther Pfahler Eisenhart**(13 January 1876 – 28 October 1965) was an American mathematician, best known today for his contributions to semi-Riemannian geometry.*Wik

**1986 Irving Reiner**was an American mathematician who (with Curtis) produced an important book on group representations.*SAU

Credits :

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*RMAT= The Renaissance Mathematicus, Thony Christie

*SAU=St Andrews Univ. Math History

*TIA = Today in Astronomy

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

## Sunday 27 October 2013

### On This Day in Math - October 27

**It is the duty of every true Muslim, man and woman, to strive after knowledge.**

Ulugh Beg [quoting the Hadith. Inscribed on his gate in Bukhara]

The 300th day of the year; 300 is a triangular number, the sum of the integers from 1 to 24. 300 is also the sum of a pair of twin primes (149 + 151). It is also the sum of ten consecutive primes, 300 = 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

**EVENTS**

**1725**Nicolaus II and Daniel Bernoulli arrived in St. Petersburg on October 27, 1725 (OS)

**In 1780**, the first U.S. astronomical expedition to record an eclipse of the sun observed the event which lasted from 11:11 am to 1:50 pm. The observers left about three weeks earlier, on 9 Oct from Harvard College, Cambridge, Mass., for Penobscot Bay, led by Samuel Williams. A boat was supplied by the Commonwealth of Massachusetts the four professors and six students. Although the U.S. was at war with Britain, the British officer in charge of Penobscot Bay permitted the expedition to land and set up equipment to observe the predicted total eclipse of the sun. The expedition was shocked to find itself outside the path of totality. They saw a thin arc of the sun instead of its complete obscuration by the moon. *TIS

**1980**The first major network crash, the four-hour collapse of the ARPANET, occurred

The ARPANET, predecessor of the modern Internet, was set up by the Department of Defense Advanced Research Projects Agency (DARPA). Initially it had linked four sites in California and Utah, and later was expanded to cover research centers across the country.

The network failure resulted from a redundant single-error detecting code that was used for transmission but not storage, and a garbage-collection algorithm for removing old messages that was not resistant to the simultaneous existence of one message with several different time stamps. The combination of the events took the network down for four hours. *CHM

**2011**

*EPL (Europhysics Letters)*went beyond Earthly limits by publishing its first ever paper submitted from space: a landmark for both European and physics-based research. Concerned with the properties of complex plasma in almost zero gravity conditions, the paper represents collaborative research of 29 individual missions performed over the last 10 years by German and Russian researchers aboard the International Space Station (ISS).

The experiments detailed in the paper were performed on the ISS in July 2010 by Alexander Alexandrovich Skvortsov and were submitted on 27 October 2011 by Skvortsov’s colleague, Sergey Alexandrovich Volkov, who remains on the ISS. IOP Blog

**BIRTHS**

**1678 Pierre Rémond de Montmort**(27 Oct 1678 in Paris, France, 7 Oct 1719 in Paris, France) was a French mathematician who wrote an important work on probability. Montmort's reputation was made by his book on probability Essay d'analyse sur les jeux de hazard which appeared in 1708. The book, which is a collection of combinatorial problems, is a systematic study of games of chance and shows that there is important mathematics in this area.

Montmort collaborated with Nicolaus(I) Bernoulli and he was also a friend of Taylor. At a time of high feelings in the Newton-Leibniz controversy it says a lot for Montmort that he could be friends with followers of both camps.

In addition to those mentioned above, Montmort corresponded with Craig, Halley, Hermann and Poleni.

Montmort was elected to be a Fellow of the Royal Society in 1715, when he was on a trip to England. The following year he was elected to the Académie Royal des Sciences. *SAU

**1728 James Cook**(27 Oct 1728; 14 Feb 1779) English seaman who was the first of the really scientific navigators. Captain Cook spent several years surveying the coasts of Labrador and Newfoundland. He observed a solar eclipse on 5 Aug 1766 near Cape Ray, Newfoundland. On the first of three expeditions into the Pacific (1768) he took Joseph Banks as the ship's botanist to study the flora and fauna discovered. (This practice of carrying a naturalist took place some 75 years before Charles Darwin's famous voyage.) Cook observed the transit of Venus on this voyage from the island of Tahiti on 3 Jun 1769. This would help scientists plot the distance between the sun to the earth. His geographical discoveries made him the most famous navigator since Magellan. He was killed by cannibal natives in Hawaii.*TIS

**1798 Heinrich Ferdinand Scherk**(27 Oct 1798 in Poznań, Poland - 4 Oct 1885 in Bremen, Germany) was a mathematician born in what is now Poland who discovered an important example of a minimal surface. Scherk discovered the third non-trivial examples of a minimal surface which appeared in his paper Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen published in Crelle's Journal. The first two examples, the catenoid and the helicoid (also called the screw surface), had been found by the Frenchman Jean Baptiste Marie Meusnier in 1776. The catenoid arises from rotating the catenary curve about a horizontal line. Scherk's result was certainly seen as a major breakthough and brought him considerable fame; two surfaces, Scherk's First Surface and Scherk's Second Surface, as they are named today, are studied in the paper. Scherk's doubly periodic surface is the first example of a complete, embedded, doubly periodic minimal surface. His minimal surfaces have recently been the basis of sculptures by the American artist Brent Collins who has based many of his works on Scherk's second minimal surface.

Another contribution by Scherk is still important today, namely his work on the distribution of the prime numbers. *SAU

**1827 Pierre-Eugène-Marcellin Berthelot**(27 Oct 1827, 18 Mar 1907 at age 79) was a French chemist and science historian and government official whose creative thought and work significantly influenced the development of chemistry in the late 19th century. He helped to found the study of thermochemistry, introduced a standard method for determining the latent heat of steam, measured hundreds of heats of reactions and coined the words exothermic and endothermic. Berthelot systematically synthesized organic compounds, including some not found in nature. His syntheses of many fundamental organic compounds helped to destroy the classical division between organic and inorganic compounds. *TIS

**1856 Ernest William Hobson**(27 Oct 1856 in Derby, England, 19 April 1933 in Cambridge, Cambridgeshire, England) wrote the first English book on the measure theory and integration of Baire, Borel and Lebesgue. *SAU

**1890 Olive Clio Hazlett**(October 27, 1890 - March 8, 1974) was an American mathematician who spent most of her career working for the University of Illinois. She mainly researched algebra, and wrote seventeen research papers on subjects such as nilpotent algebras, division algebras, modular invariants, and the arithmetic of algebras.*Wik She was the most prolific of the US-born women of her time who worked in pure mathematics and was recognized for her research accomplishments when, in 1927, she became the second US-born woman to be ranked as one of American’s leading mathematicians by her peers, a distinction marked by a “star” in American Men of Science. *Natl Museum of American History

**1915 Robert Alexander Rankin**(27 Oct 1915 in Garlieston, Wigtownshire, Scotland, 27 Jan 2001 in Glasgow, Scotland) studied at Cambrige University. His fellowship there was interrupted by his wartime work on rockets. He became Professor of Mathematics at Birmingham before moving to the professorship at Glasgow, a post he held for 27 years. His most important work was on Number Theory. He became President of the EMS in 1957 and 1978 and an honorary member in 1990. *SAU

**DEATHS**

**1449 Ulugh Beg**(22 Mar 1394- 27 Oct 1449) The only important Mongol scientist, mathematician, and the greatest astronomer of his time. His greatest interest was astronomy, and he built an observatory (begun in 1428) at Samarkand. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. After Ulugh Beg was assassinated by his son, the observatory fell to ruins by 1500, rediscovered only in 1908. Written in Arabic, his work went unread by the world's next generation of astronomers. When his tables were translated into Latin in 1665, telescopic observations had surpassed them. *TIS

1616 Johann Richter or Johannes Praetorius (1537 Jáchymov, Bohemia – 27 October 1616, Altdorf bei Nürnberg) was a Bohemian German mathematician and astronomer. From 1557 he studied at the University of Wittenberg, and from 1562 to 1569 he lived in Nuremberg. His astronomical and mathematical instruments are kept at Germanisches Nationalmuseum in Nuremberg.

In 1571 be became Professor of mathematics (astronomy) at Wittenberg where he met Valentinus Otho(Otto) and Joachim Rheticus. When Otho came to Wittenberg in 1573, he suggested to him the fraction |( \frac{355}{113}\) as an approximation to pi. Although known much earlier in the Orient, this is the first known time it was introduced in Europe.

He taught Copernicus' theory of astronomy initially as a means of eliminating the equant from Ptolemy's account, and later moving to a proto-Tychonic system.

He died in Altdorf bei Nürnberg, aged about 79. *Wik

**1845 Jean-Charles-Athanase Peltier**(22 Feb 1785, 27 Oct 1845) French physicist who discovered the Peltier effect (1834), that at the junction of two dissimilar metals an electric current will produce heat or cold, depending on the direction of current flow. In 1812, Peltier received an inheritance sufficient to retire from clockmaking and pursue a diverse interest in phrenology, anatomy, microscopy and meteorology. Peltier made a thermoelectric thermoscope to measure temperature distribution along a series of thermocouple circuits, from which he discovered the Peltier effect. Lenz succeeded in freezing water by this method. Its importance was not fully recognized until the later thermodynamic work of Kelvin. The effect is now used in devices for measuring temperature and non-compressor cooling units. *TIS

**1675 Gilles Personne de Roberval**(8 Aug 1602- 27 Oct 1675) French mathematician who developed powerful methods in the early study of integration, writing Traité des indivisibles. He computed the definite integral of sin x, worked on the cycloid and computed the arc length of a spiral. Roberval is important for his discoveries on plane curves and for his method for drawing the tangent to a curve, already suggested by Torricelli. This method of drawing tangents makes Roberval the founder of kinematic geometry. In 1669 he invented the Roberval balance with an articulated parallelogram is now almost universally used for weighing scales of the balance type. He studied the vacuum and designed apparatus which was used by Pascal in his experiments and also worked in cartography. *TIS

**1968 Lise Meitner**(7 Nov 1878, 27 Oct 1968)Austrian physicist who shared the Enrico Fermi Award (1966) with the chemists Otto Hahn and Fritz Strassmann for their joint research beginning in 1934 that led to the discovery of uranium fission. She refused to work on the atom bomb. In 1917, with Hahn, she had discovered the new radioactive element protactinium. She was the first to describe the emission of Auger electrons. In 1935, she found evidence of four other radioactive elements corresponding to atomic numbers 93-96. In 1938, she was forced to leave Nazi Germany, and went to a post in Sweden. Her other work in the field of nuclear physics includes study of beta rays, and study of the three main disintegration series. Later, she used the cyclotron as a tool. *TIS

**1980 John Hasbrouck Van Vleck**(13 Mar 1899, 27 Oct 1980) was an American physicist and mathematician who shared the Nobel Prize for Physics in 1977 with Philip W. Anderson and Sir Nevill F. Mott. The prize honoured Van Vleck's contributions to the understanding of the behaviour of electrons in magnetic, noncrystalline solid materials. *TIS

**1999 Robert L. Mills**(15 Apr 1927, 27 Oct 1999)American physicist who shared the 1980 Rumford Premium Prize with his colleague Chen Ning Yang for their "development of a generalized gauge invariant field theory" in 1954. They proposed a tensor equation for what are now called Yang-Mills fields. Their mathematical work was aimed at understanding the strong interaction holding together nucleons in atomic nuclei. They constructed a more generalized view of electromagnetism, thus Maxwell's Equations can be derived as a special case from their tensor equation. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories. *TIS

Credits :

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*RMAT= The Renaissance Mathematicus, Thony Christie

*SAU=St Andrews Univ. Math History

*TIA = Today in Astronomy

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

## Saturday 26 October 2013

### On This Day in Math - October 26

**There are no foolish questions and no man becomes a fool until he has stopped asking questions.**

Charles P Steinmetz

The 299th day of the year; If a cubic cake was cut with 12 straight cuts, it can produce a maximum of 299 pieces.... a good day to "let 'em eat cake."

**1676**Newton, through the intermediary of Oldenburg, wrote Leibniz concerning his work on the calculus. An anagram contained the statement of the problem of integrating differential equations. *VFR

1738 Theophilus Grew, Mathematician advertises his availability for all manner of mathematical instruction in the Philadelphia Gazette.

"Forasmuch as Mathematical Learning is (and has been in all Ages) promoted in most Parts of the World especially in all great Towns, and generally pursued by the Gentry and those of the first Rank, as a necessary Qualification; it is to be hoped that this flourishing City will follow the Example and give it such Encouragement as it justly deserves.Grew would go on to become the first professor of mathematics at the Univ of Pennsylvania when it was founded. *Natl. Archives

In order to which there will be taught this Winter, over against Mr. James Steel’s in Second-Street, Philadelphia; Reading, Writing, Vulgar Arithmetick, Decimal Arithmetick, Accompts, Euclid’s Elements, Practical Geometry, Mensuration, Gauging, Surveying, Algebra, Trigonometry, Geography, Navigation, Astronomy, Dialing, Projection of the Sphere, the Use of Globes, Maps, Quadrants, Scales, sliding Rules, and all other Instruments for the Mathematical Service, by Theophilus Grew, Mathematician."

1818 Thomas Jefferson writes Nathaniel Bowditch to offer him the Math Professorship at the newly forming University of Virginia

I have stated that where men of the 1st. order of science in their line can be found in our country, we shall give them a willing preference. we are satisfied that we can get from no country a professor of higher qualifications than yourself for our Mathematical department, and we entertain the hope and with great anxiety that you will accept of it. the house for that Professorship will be ready at midsummer next or soon after, when we should wish that school to be opened. I know the prejudices of every state against the climates of all those South of itself: but i know also that the candid traveller advancing Southwardly, to a certain degree at least, sees that they are more prejudices, and that the real advantages of climate are in the middle & temperate states, and especially when above their tide waters.*Letters of Thomas Jefferson, http://etext.lib.virginia.edu

**1843**John T Graves replies to Hamilton about the invention of Quaternions,

"There is something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties.",

"If with your alchemy you can create three pounds of gold, why should you stop there?

Graves is credited by Hamilton with being a critical inspiration in the Quaternions, and would quickly go on to create the "Octaves", an eight dimensional normed division algebra. Why should you stop there indeed? *Joan Baez Rankin Lecture of September 17, 2008 Glascow

**1847**William Whewell wrote to Aubrey De Vere expressing dismay at the influence of Carlyle's pessimism among his friends and in society. *@GalileosBalls, Twitter

1896 Comptes Rendus publishes, "Extension of the Reimann-Roch Theorem to Algebraic Surfaces. A note by M. M. Noether, presented by M. Hermite *Mathematical Intellignecer vol 8 #4

**1893**Karl Pearson’s ﬁrst statistical publication. *VFR In Pearson' s first published statistical paper of 26 October 1893, he introduced the method of moments as a means of curve fitting asymmetrical distributions. One of his aims in developing the method of moments was to provide a general method for determining the values of the parameters of a frequency distribution. *StatProb web site

**1960**Saga, a silent shoot-em-up Western playlet made on the TX-0 computer, was run on CBS' special for MIT's 100th anniversary. The TX-0 was the first general purpose transistorized computer. The program for Saga comprised 4,096 words of magnetic core storage. The 13,000 lines of code choreographed the movements of each object. A line of direction was written for each action, even if it went wrong. This led to the high point of the show where sheriff put his gun in the holster of the robber resulting in a never ending loop.

Doug Ross explained the rule-based diagram: If the robber drank from alcohol, his judgement would start to decline, but the program would remain logical.*CHM

1846 Lewis Boss (26 Oct 1846; 12 Oct 1912) American astronomer best known for his compilation of two catalogues of stars (1910, 1937). In 1882 he led an expedition to Chile to observe a transit of Venus. About 1895 Boss began to plan a general catalog of stars, giving their positions and motions. After 1906, the project had support from the Carnegie Institution, Washington, D.C. With an enlarged staff he observed the northern stars from Albany and the southern stars from Argentina. With the new data, he corrected catalogs that had been compiled in the past, and in 1910 he published the Preliminary General Catalogue of 6,188 Stars for the Epoch 1900. The work unfinished upon his death was completed by his son Benjamin in 1937 (General Catalogue of 33,342 Stars for the Epoch 1950, 5 vol.)*TIS

**1849 Georg Frobenius**(26 Oct 1849; 3 Aug 1917) German mathematician who made major contributions to group theory, especially the concept of abstract groups (with Ludwig Stickleberger) and the theory of finite groups of linear substitutions (with Issai Schur), that later found important uses in the theory of finite groups as it applies to quantum mechanics. He also contributed to means of solving linear homogenous differential equations. The fact so many of Frobenius's papers read like present day text-books on the topics which he studied is a clear indication of the importance that his work, in many different areas, has had in shaping the mathematics which is studied today.*TIS

**1877 Max Mason**(26 Oct 1877; 23 Mar 1961) American mathematical physicist, educator, and science administrator. During World War I he invented several devices for submarine detection - several generations of the Navy's "M," or multiple-tube, passive submarine sensors. This apparatus focused sound to ascertain its source. To determine the direction from which the sound came, the operator needed only to seek the maximum output on his earphones by turning a dial. The final device had a range of 3 miles. Mason's special interest and contributions lay in mathematics (differential equations, calculus of variations), physics (electromagnetic theory), invention (acoustical compensators, submarine-detection devices), and the administration of universities and foundations. *TIS

**1885 Niels Erik Norlund**(26 Oct 1885 in Slagelse, near Soro, Sjaelland, Denmark - 4 July 1981 in Copenhagen, Denmark) In 1907 he was awarded a gold medal for an essay on continued fractions and his resulting two publications were in 1908: Sur les différences réciproques; and Sur la convergence des fractions continues both published in Comptes Rendus de l'Academie des Sciences. These publications in the most prestigious French journal earned Norlund an international reputation despite still being an undergraduate. In the summer of 1910 he earned a Master's degree in astronomy and in October of that year he successfully defended his doctoral thesis in mathematics Bidrag til de lineaere differentialligningers Theori. In the same year he published the 100-page paper Fractions continues et différences réciproques as well as Sur les fractions continues d'interpolation, a paper on Halley's comet, and an obituary of his teacher Thorvald Thiele. Norlund's sister Margrethe married Niels Bohr whose brother, Harald, was also an outstanding mathematician. In 1955 Norland reached retirement age. That mathematics was his first love now became clear, for once he gave up the responsibilities of the Geodesic Institute he returned to mathematics research. He published Hypergeometric functions in 1955 which was reviewed by Arthur Erdélyi, "This is one of those rare papers in which sound mathematics goes hand in hand with excellent exposition and style; and the reader is both instructed and delighted. It is likely to become the standard memoir on the generalized hypergeometric series ... " The paper Sur les fonctions hypergéométriques d'ordre supérieur (1956) gives a very full, rigorous and classical treatment of some integrals from generalized hypergeometric function theory.*SAU

**1902 Henrietta Hill Swope**(26 October 1902; Saint Louis, Missouri - 24 November 1980; Pasadena, California)was an American astronomer. She was the eldest child of Gerard and Mary Dayton (Hill) Swope; her mother was the daughter of Thomas Hill, president of Harvard University, 1862-1868. She received her A.B. from Barnard College in 1926 and her A.M. from Radcliffe College in 1928. In 1936, while assistant at the Harvard Observatory (1928-1942), she was a member of the expedition sent jointly by the Harvard Observatory and the Massachusetts Institute of Technology to study the solar eclipse in Soviet Central Asia. During World War II she was staff member of the M.I.T. Radiation Laboratory and then served as a mathematician in the Hydrographic Office of the U.S. Department of the Navy. From 1947 to 1952 she taught astronomy at Barnard College and in 1952 was appointed assistant, later research fellow, at the Mt. Wilson and Palomar Observatories in California. After her retirement in 1968, she continued to work at the Observatories.

HHS was a member of the American Astronomical Society; she received the AAS Annie Jump Cannon Prize in 1968 for her research on photometry and variable stars. She was responsible for developing a new yardstick for measuring the universe: calibrating distance by determining the brightness of stars. She received the Distinguished Alumna Award of Barnard College in 1975 and the Barnard Medal of Distinction in 1980.

The Swope Telescope at the Las Campanas Observatory in Chile is named in her honor, as is asteroid 2168 Swope.

**1911 Shiing-shen Chern**(26 Oct 1911; 3 Dec 2004) Chinese-American mathematician and educator whose researches in differential geometry include the development of the Chern characteristic classes in fibre spaces, which play a major role in mathematics and in mathematical physics. "When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point. Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people. Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern." *TIS

1930 Walter Feit (26 Oct 1930 in Vienna, Austria - 29 July 2004 in Branford, Connecticut, USA) was an Austrian mathematician who (with John Thompson) proved one of the most important theorems about finite simple groups. In 1990 his 60th birthday was celebrated with an 'International Symposium on the Inverse Galois Problem' held in Oxford. His retirement from Yale in October 2003 was marked with the holding of a 'Conference on Groups, Representations and Galois Theory' in his honour. Feit died after a long illness at the Connecticut Hospice in Branford, Connecticut, USA. A memorial service was held on Sunday 10 October 2004 at the New Haven Lawn Club, New Haven, Connecticut. *SAU

**1817 Aida Yasuaki**was a Japanese mathematician who published about 2000 works. Aida compiled Sampo tensi shinan which appeared in 1788. It is a book of geometry problems, developing formulae for ellipses, spheres, circles etc. Aida explained the use of algebraic expressions and the construction of equations. He also worked on number theory and simplified continued fraction methods due to Seki. *SAU

**1923 Charles Proteus Steinmetz**(9 Apr 1865- 26 Oct 1923) German-born American inventor and electrical engineer whose theories and mathematical analysis of alternating current systems helped establish them as the preferred form of electrical energy in the United States, and throughout the world. In 1893, Steinmetz joined the newly organized General Electric Company where he was an engineer then consultant until his death. His early research on hysteresis (loss of power due to magnetic resistance) led him to study alternating current, which could eliminate hysteresis loss in motors. He did extensive new work on the theory of a.c. for electrical engineers to use. His last research was on lightning, and its threat to the new AC power lines. He was responsible for the expansion of the electric power industry in the U.S. *TIS

**1968 Sergei Natanovich Bernstein**(March 5, 1880 – October 26, 1968) was a Russian and Soviet mathematician. His doctoral dissertation, submitted in 1904 to the Sorbonne, solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. Later, he published numerous works on Probability theory, Constructive function theory, and mathematical foundations of genetics. From 1906 until 1933, Bernstein was a member of the Kharkov Mathematical Society. *Wik

1970 Marcel Gilles Jozef Minnaert (12 Feb 1893; 26 Oct 1970 at age 77)

Flemish astronomer and solar physicist who was one of the pioneering solar researchers during the first half of the 20th century. Applying solar spectrophotometry, he was one of the first to make quantitative measurements of the intensity distribution inside Fraunhofer lines, and interpret from them information about the outer solar layers. His range of study also included comets, nebulae and lunar photometry. During the time he was director of the observatory at the University of Utrecht, (1937-1963) he created a modern astronomical institute to study solar and stellar spectra with resources including a solar telescope, spectrograph, photometer, and mechanical workshop. Minnaert also maintained a strong interest in the education of physics teachers, and as a univeristy professor gave clear, enthusiastic and well-prepared lectures. *TIS

**1983 Alfred Tarski**(14 Jan 1902, 26 Oct 1983) Polish-born American mathematician and logician who made important studies of general algebra, measure theory, mathematical logic, set theory, and metamathematics. Formal scientific languages can be subjected to more thorough study by the semantic method that he developed. He worked on model theory, mathematical decision problems and with universal algebra. He produced axioms for "logical consequence", worked on deductive systems, the algebra of logic and the theory of definability. Group theorists study 'Tarski monsters', infinite groups whose existence seems intuitively impossible. *TIS

**1984 Mark Kac**(3 Aug 1914 in Krzemieniec, Poland, Russian Empire - 26 Oct 1984 in California, USA) pioneered the modern development of mathematical probability, in particular its applications to statistical physics. The method of quantization now in use involves the Feynman-Kac path integral, named after Richard Feynman and Mark Kac. He published a classic text Statistical Independence in Probability, Analysis and Number Theory in 1959. To many Kac will be remembered best for a paper he wrote for the American Mathematical Monthly in 1966. This is the famous paper Can One Hear the Shape of a Drum? and Kac received the Chauvenet Prize from the Mathematical Association of America in 1968 for the, "most outstanding expository article on a mathematical topic by a member of the Association." *SAU

1998 Kenkichi Iwasawa (11 Sept 1917 in Shinshuku-mura (near Kiryu), Gumma Prefecture, Japan - 26 Oct 1998 in Tokyo, Japan ) In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field. This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves. "it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem." *SAU

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 & Campbe

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