## Sunday, 8 December 2013

### Antiparallels, an Overlooked HS Beauty

I would think it is pretty fundamental in typical HS classrooms that students recognize that a line parallel to one of the sides of a triangle will cut the other two sides in a pair of angles which are congruent to the angles formed at the third side. Eventually they can prove that the triangle formed by the parallel line forms a triangle with the two sides similar to the original triangle.

Almost none of them, and perhaps very few of their teachers, know that there is a second type of line which can be drawn to cut the two sides which will also form a similar triangle, and thus must also form angles congruent to the two base angle of the original triangle. Its called the anti-parallel now, but it used to be called a subcontrary line, at least by Apollonius.

There are several nice ways to produce an antiparallel in a triangle. A nice general way is to use the two vertices of one leg and a point on one of the other two legs to construct a circle. The circle will then cut the other leg in a fourth point which is the other vertex of the antiparallel. These four point are the vertices of a cyclic quadrilateral, for which the opposite angles are supplementary. This makes it easy to see that the antiparallel forms angles on one leg congruent to the original angle on the other leg.
A second way is to draw an altitude from two of the vertices to the opposite sides. The segment connecting the feet of these two altitudes is also antiparallel to the third side.
*image from Wolfram Mathworld

If you construct the circumcircle of the triangle, the tangent at the vertex opposite a side will be anti-parallel to that side. (This doesn't strike me as a simple proof, but I may be overlooking something. I often do. If you have a simple proof, high school level for example, I would love to see and share it.)

Just as the median bisects all lines parallel to the base, its reflection in the angle bisector (called the symmedian) will bisect each anti-parallel.

The antiparallel shows up as the solution to an optimization problem that was first proved by Giovanni Fagnano in 1775: For a given acute triangle determine the inscribed triangle of minimal perimeter. Turns out the answer is the triangle formed by the three anti-parallels connecting the feet of the three altitudes, called the orthic triangle.

For slightly more advanced students who have been exposed to cones it is constructive to point out that for an oblique circular cone, (one in which the axis is not perpendicular to the base; and many students graduate from HS without ever having been made aware that such types of cones exist, much less those whose base is non-circular) there is more than one plane which will cut a circle. A cutting plane parallel to the base is one type, and of course by now you suspect that the other type is a plane anti-parallel to the base.
*image from Paramanand's Math Notes

Maybe soon I'll write about the anti-CENTER.

Addendum: After a comment by 1SAEED9, I realized that the proof that the tangent at the vertex opposite a side is antiparallel to that side.

It is easy to see that angles DBA and BCA both subtend the same arc, and thus are the same measure. By using the fact that CBA, DBA and EBD add up to 180 degrees, and the three interior angles of the triangle CBA, CAB, and BCA also add up to 180 degrees. Since BCA and DBA are congruent, when we subtract these from each side, and remove CBA from both sides we are left with the fact that EBD must be congruent to CAB. We can repeat this process on the opposite angle and we are done. Easier than I imagined.

## Monday, 11 November 2013

### Vinculum is a Collective Noun

I'm not a pedant. I'm really not... really!

Ok let me explain. Murray Bourne, who writes a really nice math blog called Square Circle Z, as part of his Interactive Math site, (Both are excellent, if you haven't been there, go there) wrote a tweet a couple of days ago that got me started. He wrote, "A 'vinculum' is a horizontal line indicating grouping. E.g. over the '14' in 14/99=0.141414... "

I could have shook my head a little on gone on to the next post, but I like Murray, and think he is probably an excellent teacher. So I sent a brief quibble, which on twitter ran to about three tweets. But I think he missed my point, or perhaps he just didn't think the distinction I was trying to make was important. So here I am trying to tell the world to change to my way of thinking, and I hope with more than 140 characters, I can explain what I mean, and why I think the distinction is important.
The fact is that the overbar in the notation of repeating decimals is the only reference students have for vinculum. I will suggest (encourage/plead) that teachers add the common uses of parentheses and brackets as part of their description of a vinculum .
Many US teachers know of no other representation of repeating decimal fractions, yet they seem to have been the last application of the bar, and seem not to have occurred until after 1930 in the US. In F. Cajori's A History of Mathematical Notations (1929) he points out two forms of marking repeating sequences in decimals but does not mention the overbar. Cajori credits John Marsh [Decimal Arithmetic Made Perfect, (London, 1742)] with being the first to use a symbol to indicate the repeat sequence. Marsh sometimes placed a single dot over the first number in the repeat sequence, and sometimes placed one on the first and last.

This was one of the most frequent in the early arithmetics in the US, possibly due to the fact that many of them were by British authors,  or near verbatim copies of their books. John Bonnycastle and other British came early to the country to work in the early universities.

Like many terms of mathematical interest, vinculum(vincula) is a term that used to be better known. It seems many teachers have only a very limited knowledge of the history of even common arithmetic notations.   I've previously wrote responses to a teacher who gave me flack because I used "reduce", leading me to write this blog, "On Reducing Fractions". And another complained when I suggested that the number one, has been, and could be, labeled a prime leading me to write, "One is Prime if we Wish it to Be."

As the title says, vinculum is a collective noun, like truck, or variable. My old Dodge is a truck. It's not the only kind of truck. Some people have new trucks. Some people even have Ford trucks. I know; but what can you say to them. It's not that I'm prejudice. My own sister drives a Ford and I still love her like a, well, like a sister. And x is a variable, but it's not the only variable, and it is not always a variable, sometimes it is just a letter at the beginning of xenophobe. And if you had a student who argued that y can't be the variable because x is the variable, you would want to give them a more complete explanation.

The horizontal bar above a repeating decimal, such as $.\overline{14}$ is an example of a vinculum. It is now almost the only term that people use that term for, I think because they think it is a name for the bar, rather than a description of it's role in that situation. Horizontal bars were once commonly used beneath repeating decimals, and in fact beneath algebraic expressions in the same way we use grouping symbols today.
In "The Constructive Arithmetic" by James A Christie (1865) he writes, "The bracket { }, or [ ], or horizontal bar (such as sometimes separates the numerator of a fraction from its denominator,) is sometimes employed as a vinculum." Later he writes :

His interpretation of vinculum is a little unusual, as it is generally interpreted as something like binder. One dictionaries etymology gives "from vincire, vinctum, to bind." I have read that it was the name used frequently for a hobble for the legs of cattle in the field to keep them from wandering off. It was something like manacles and meant to allow the animal to move but keep it from moving quickly.
In "A Treatise on Arithmetic: Through which the Entire Science Can be Most Expeditiously and Perfectly Learned, Without the Aid of a Teacher." By Noble Heath he gives :

On another web site I have written, "In the same year as the 29th NCTM yearbook(1964), Irving Adler obtained a copyright for A New Look At Arithmetic, and on page 220 he writes, 'To indicate a repeating decimal with a minimum of writing, it is customary to write only enough decimal places to include the repeating part once, and to identify the repeating part by underlining it. Thus the repeating decimal for $\frac{211}{990}$  is therefore represented by $.\underline{213}$. '. It is worth mentioning that William Oughtred, the 16th Century mathematician indicated all decimals by underlining. "

Another example, or rather a hybrid of two of the former, also appeared in a book with a 1964 copyright. A A Klaf's Arithmetic Refresher was published a few years after his death by his family. The book is written in a question and answer style somewhat reminiscent of the classic dialogs of antiquity. On page 188 it asks, "How are recurring, circulation, or repeating decimals denoted?" It then goes on to answer, "b) by dots placed over the first and last figures of the recurring group." This is described exactly like the more common earlier usage, but the figure that follows includes dots, and then an arc above them, similar to what I have shown here. Similar arcs were used over groups of three numbers to indicate the periods (thousands, millions, etc) in some early use of Hindu-Arabic numerals. Gerber(980), who later became Pope Sylvester, referred to them as "Pythagorean Arcs."

A popular author of arithmetics in the US in the 19th century was Charles Davies. He was one of the original instructors at the US Military at West Point. In his New University Arithmetic (1860) he uses yet a different type of vinculum than all the others I have mentioned. Davies sets off the repeating digits with a pair of single quotes, so 1/6 would be written .1'6'.

The more general definition may be slipping from use, but I think it is worthwhile to preserve the distinction.  When a symbol is used to bind together other numbers or operations, it is acting as a vincula, whether it is the fraction bar, $\frac{a}{b}$, or the diagonal solidus between fractions, a/b, a parenthesis ln[4{3+2(x+y)}] or brackets.  And when I type two dollar signs around an expression in Latex to make it print it as pretty math, those dollar signs form a vinculum to bind that expression together so that the computer knows, "This is math, print it using the math library I mentioned in the header."

ADDENDUM:  In the comments, Murray writes with about a problem many teachers have encountered, how do you write repeating decimals on a typewriter or word processor if you don't have $LaTex$ or an equation editor (often not available to middle school teachers and others who teach repeating decimals)? He suggest using a square bracket vinculum (in the manner of James A Christie) to set off the repeating part.  It seems a wonderful idea.  They are distinct, and in  this usage, not easily confused with other potential uses of brackets at that (or any other?) level.  So 1/11 would be .[09], and 1/6 would be .1[6]  and no special typesetting needed.  I think if middle school teachers all over the country started using this it might force the higher school teachers to adapt, or by that time, maybe a different notation wouldn't be a problem for the students.  Being told to switch, they would begin to realize that the notations used in math are matters of choice, after all, we didn't always use = for equal.  Well after I received this suggestion from Murray, one of my brilliant ex students (brilliant and my ex-student, no causative effect suggested), Jacob Coakwerll told me that in Russia, where he lived for an extended period of time, the repeating part is included in parenthesis, so 7/12=  0,58(3)  where the comma is used for the decimal seperatrix.

I also just noticed that some of the "old" symbols mentioned here may not be extinct.    In answer to a question on Yahoo Answers asking, "what is the name of the repeating sign over decimals?"  The answers included, "I don't know if it has a name, i just call it a dot, cos my teacher taught me to put a dot over each number that repeats."    Another seemed to suggest that something like Murray's practice was already in use, "It's just called a bar. Sometimes you will see (6) instead of a bar above or underneath the number."  A Wikipedia article suggested that the parenthetical use is mostly in Europe.

I would love for folks in different area around the world to write and tell me how they do repeating decimals. I sent a twitter question out and here are some of the responses:
Thony Christie ‏@rmathematicus England, "Bar over the repeat period and a period after the last digit."  $0.\overline{23}.$
MathsEnVideo ‏@MathsEnVideo " In France: same as in the USA or with points over the period's digits."  $0.\dot{2}\dot{3}$
Dong Suk Smith, an ex-student of Korean origin remembers that there they use a dot over the repeating period digits.
A teaching friend who has lived and retired in Japan writes that his wife has never seen the over-line and that the Japanese seem to use a repetition of the repeating period followed by an ellipsis.

## Friday, 1 November 2013

### Mobius Double Cross

As a grandfather, I always love being able to take time over the holidays to share entertaining math enrichments with the grandkids. Last Christmas I showed them this one and it was a big hit
This, to me, is the greatest Mobius related activity I have ever seen.  I wrote about it briefly as part of a longer blog, but wanted to focus one on just this neat activity. (I have also included a link at the bottom to a nice Matt Parker video with a view others.   I got this from an Ivars Peterson article in the New York Times
Don't read the article until after you have tried it, He offers teasers of what the outcome could be.

Start by cutting out a cross of paper.  Make the sides wide enough to do some cutting, one of them into thirds (sort of).

Now take ends of one cross and give them the standard half-twist to make a Mobius strip with a cross piece hanging on.  Now take the other crossing pair and fold them away from the Mobius loop to make a regular (non-twisted) loop.  It should look sort of like a twisted figure eight.

Here is an image of what it may look like from one of his pages.

Now draw a line trisecting the Mobius branch.  One line 1/3 of the way across the page should loop back around the loop and eventually make something like three paths on both sides of the strip.
Now cut along the trisected loop, then bisect the non-Mobius loop.

Shake out all the twists and turns to be amazed.

A while after I wrote this, I came upon a video of a talk by Matt Parker in which he includes several demonstrations kids (of all ages) would enjoy using Mobius strips of different numbers of twists, including zero twists.

So if you are looking for a way to share your love of math over the holidays with young people, you could work up a nice routine with some of these.
Enjoy!

## Thursday, 31 October 2013

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

"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."

## 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 ab 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=ab 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) = 21/2 so sqrt(2)^sqrt(2) = 21/2sqrt(2) and using the product of powers we can get 2sqrt(2)/2. By using sqrt(2)/2 = 1/sqrt(2) we finally arrive at sqrt(2)^sqrt(2)= 21/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 ii 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 aa has a certain property (rational for instance)  . I think it would be nice to have a collection of irrational pairs a,b such that ab 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..

### 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

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 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

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.
~Jean d'Alembert [advice to those who questioned the calculus](probably also great for students struggling with mathematics at any level)

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 x2 and x3,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
1897 Edwin James George Pitman was born in Melbourne on 29 October 1897 and died at Kingston near Hobart on 21 July 1993.  In 1920 he completed the degree course and graduated B.A. (1921), B.Sc. (1922) and M.A. (1923). In the meantime he was appointed Acting Professor of Mathematics at Canterbury College, University of New Zealand (1922-23). He returned to Australia when appointed Tutor in Mathematics and Physics at Trinity and Ormond Colleges and Part-time Lecturer in Physics at the University of Melbourne (1924-25). In 1926 Pitman was appointed Professor of Mathematics at the University of Tasmania, a position he held until his retirement in 1962.
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,

~Augustus De Morgan

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