Showing posts with label lewis Carroll. Show all posts
Showing posts with label lewis Carroll. Show all posts

Saturday, 18 June 2011

On This Day in Math - June 18


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


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

Image from Wikipedia.


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



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

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

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


BIRTHS
1799 William Lassell was a wealthy amateur English astronomer. He set up an observatory at Starfield, near Liverpool. England, He built his own 24" diameter telescope, and devised steam-driven equipment for grinding an polishing the speculum metal mirror. This telescope was the first of its size to be mounted "equitorially" to allow easy tracking of the stars. He discovered Triton, a moon of Neptune, and Ariel and Umbriel, satellites of Uranus. Later, Lassell built a 48" diameter telescope with th same design and took it to Malta for observations with clearer skies.*TIS
1818 Pietro Angelo Secchi (18 Jun 1818, 26 Feb 1878 at age 59) Italian Jesuit priest and astrophysicist, who made the first survey of the spectra of over 4000 stars and suggested that stars be classified according to their spectral type. He studied the planets, especially Jupiter, which he discovered was composed of gasses. Secchi studied the dark lines which join the two hemispheres of Mars; he called them canals as if they where the works of living beings. (These studies were later continued by Schiaparelli.) Beyond astronomy, his interests ranged from archaeology to geodesy, from geophysics to meteorology. He also invented a meteorograph, an automated device for recording barometric pressure, temperature, wind direction and velocity, and rainfall.*TIS
1858 Andrew Russell Forsyth born in Glasgow, Scotland. He studied at Liverpool College and was tutored by Richard Pendlebury before entering Trinity College, Cambridge, graduating senior wrangler in 1881.[1] He was elected a fellow of Trinity and then appointed to the chair of mathematics at the University of Liverpool at the age of 24. He returned to Cambridge as a lecturer in 1884 and became Sadleirian Professor of Pure Mathematics in 1895. *Wik 
In 1893 he published Theory of functions of a complex variable which had such an impact at Cambridge that function theory dominated there for many years. Whittaker writes... that this text:-
... had a greater influence on British mathematics than any work since Newton's Principia.
However the reputation of the book outside Britain was not high. In fact this is not surprising since the whole thrust of the book was to bring the great advances of Continental mathematics to Cambridge which Forsyth rightly saw as living in the past. He was well equipped to undertake this task for he traveled widely and, being a good linguist, was able to appreciate the advances made by authors writing in French and German.
On Cayley's death Forsyth was appointed to his chair in 1895 becoming the Sadleirian professor of Pure Mathematics. However his preference for technical mastery rather than rigorous analysis meant that he failed to inspire future pure mathematicians. In fact one would have to say that Forsyth was unlucky, for although he saw the importance of Continental mathematics, at the same time his greatest strengths lay in his ability to handle complex formulae. He therefore excelled at precisely the style of mathematics which he himself campaigned successfully to replace at Cambridge.

1884  Charles Weatherburn worked on vector analysis and differential geometry.*SAU

1884 Frieda Nugel was a German mathematician who was one of the first women to receive a doctorate in Germany *SAU 

1913  Oswald Teichmüller's main contribution is in the area of geometric function theory.*SAU

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


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



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


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


Credits:
*VFR = V Frederick Rickey, USMA
*TIS= Today in Science History
*Wik = Wikipedia
*SAU=St Andrews Univ. Math History

Monday, 7 February 2011

Follow up to Sum of Squares Problem

A shiny "Attaboy" award to Gasstationwihtoutpumps for the correct answer to both Lewis Carroll Problems. 

Herer is his solution again. 


The first result is moderately interesting:
(a+b)^2 + (a-b)^2 = 2 (a^2 + b^2)

But the second one is then trivial
(a+b)^2 + (a-b)^2 + a^2 + b^2 = 3 (a^2 + b^2)

Do you have some extra constraints for the second problem to make it non-trivial?
 

For most folks of my generation, raised on lots and lots of factoring problems, the first problem was one of a pair that jumped off the pages of algebra books so often they were almost memorized.  The sum and difference of (x+y)2 and (x-y)2 produce the  results 2( x2+y2) and 4xy; both of which seemed to occur in textbook problems of my youth.  
Had Gas... not already told me he had always had problems with memory I would think he had grown up on the same factoring fare and had also memorized them.  Perhaps he is more like Dodgeson (Carroll) who apparently wrote about re-discovering this in his diary "towards the end of his life" as he too did not recall it.  As a puzzle maker, he probably thought the addition of a third multiple would disguise a common factoring and it was this second problem that appeared in his "Pillow Problems".

 Since my own memory also is very fragile, I was fortunate to find the information in Edward Wakeling's "Lewis Carroll's Games and Puzzles". 
Carroll also included a somewhat more difficult probability problem in his "Pillow Problems".  Since I can't make the first problem more difficult for Gas.... I will give the advanced version of this one. 

An urn holds B Black and W White balls.  One is selected at Random and placed in a bag without your knowledge.  Then a White ball is added to this bag.  A random draw from this bag produces a white ball.  What is the probabilty the original ball was White?

Sunday, 6 February 2011

Sums of Squares Problems

Now even my ex-students are trying to hang me up with "Pythagorean-Like" Problems.  Today I got these two in an email. Since the problems date back to Lewis Carroll (whom  my student's assume I must  have gone to school with since I am so very old).

 I knew the problems and the solutions so I will leave them for now as problems for the reader, and post a solution in a day or two; hopefully with a litlte of the history. 
They are suitable for HS level so everyone is invited. Feel free to submit your solutions.

For any two positive integers X and Y, twice the sum of their squares can be expressed as the sum of two squares.  For example 2(42 + 72) =    2(16+49) = 2(65) = 130..but 130 = (112+32)   

And the followup??? Prove that three times the sum of two squares can be expressed as the sum of four squares. 

Enjoy

Tuesday, 19 January 2010

Lewis Carroll, Happy Birthday, and RIP



January was the month in which Lewis Carroll was both Born (27th) and Died (the 14th)

It seems a fitting time to remember some of my favorite (not all true) stories about him.
Carroll was of course the pseudonym by which he wrote, but in his day to day life he was an instructor of mathematics at Oxford by the name of Charles Lutwidge Dodgeson (Lewis Carroll is an alteration of the Latinization of Charles into Carroll, and the replacement of Lutwidge with Lewis). He was a good (but not great) mathematician, but in the way of the world, he is remembered most for his children's stories... and had he not written them, he would probably be remembered for his photography... And if he had avoided that also... Maybe people would know he was a mathematician, but probably not; how many people on the street have heard of Euler?.

An interesting, but seemingly false, story circulated about a gift of a book on determinants to the Queen of England by Lewis Carroll. Here is the version as it is told on the Mathworld page.
=========================
Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author's next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author's other books, and was sent one of Dodgson's mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson).
==============================
Ok, but still a neat story...

I love this letter, written by Dodgeson to a young man named Wilton Cox.
"Honoured Sir,
Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.

If x and y are each equal to 1, it is plain that

2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.

Hence 2 × (x2 - y2) = 5 × (x - y).

Now divide each side of this equation by (x - y).

Then 2 × (x + y) = 5.

But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll."

Another interest of Dodgson's was the analysis of tennis tournaments:

"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself." Carroll set out the guidelines for a seeding system well before the good folk at Wimbledon ever thought of it. "Good on ya" as they say over here.

And from a letter he wrote in 1868 with suggestions for essentials of math instruction



Here are links to several other blogs I have written that involve Lewis Carroll in some way..
A Brief History of Logic Diagrams

Searching for Snarks

and Two times Two is Five Enjoy...

Sunday, 26 October 2008

Searching for Snarks




The happy band of mathematical warriors above joined me on Thursday night as we set out to the Center for Mathematical Sciences at Cambridge. Our quest was to discover all that Emeritus Gresham College Professor of Geometry Robin Wilson might know about "Lewis Caroll in Numberland." The lecture, which can only be described as "math-lite" was entertaining none the less, and made even better by the good companions who joined me.

Wilson is also, not by coincidence, the author of a book on the topic that is soon to be released in the US (and can be purchased in advance from Amazon at a healthy discount) entitled, Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life.

Wilson explained that if Dodgeson (the real name of Lewis Carroll) had not written the "Alice" stories for which he is so well remembered, he might well be remembered for being one of the pioneering child photographers of the 19th century. And if he had not done either, he might be remembered as an accomplished mathematician and teacher who made contributions in the areas of Logic, algebra, geometry, and the mathematics of elections. Wilson points out that:
"Yet another interest of his was the study of voting patterns. Some of his recommendations were adopted in England, such as the rule that allows no results to be announced until all the voting booths have closed. Others, such as his various methods of proportional representation, were not. As the philosopher Sir Michael Dummett later remarked:

It is a matter for the deepest regret that Dodgson never completed the book he planned to write on this subject. Such was the lucidity of his exposition and mastery of this topic that it seems possible that, had he published it, the political history of Britain would have been significantly different."

He also credits Carroll with the invention of the modern method of seeding tennis matches:
Another interest of Dodgson's was the analysis of tennis tournaments:"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself.
Let us take sixteen players, for example, ranked in order of merit, and let us organise a tournament with 1 playing 2, 3 playing 4, and so on. Then the winners of the first round will be 1, 3, 5, and so on; those of the second round will be 1, 5, 9 and 13; the final will then be won by player 1, defeating player 9 who wins the second prize but actually started in the lower half of the ranking.
To avoid this difficulty, he managed to devise a method for re-scheduling all the rounds so that the first three prizes go to the best three players, which presaged the present system of seeding.

For a brief discription of some of his work in Logic diagrams see My page here and for A different version of the Wilson's talk on Lewis Carroll given at Gresham College, look here

Tuesday, 26 August 2008

two times two is five??? by Lewis Carroll


I recently came across an example of Carroll's cross between wit and logic in a 2008 Gresham lecture by Robin Wilson:



Honoured Sir,



Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.


If x and y are each equal to 1, it is plain that


2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.


Hence 2 × (x2 - y2) = 5 × (x - y).


Now divide each side of this equation by (x - y).


Then 2 × (x + y) = 5.


But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.


Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.



I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll.