Charles Lutwidge Dodgson

This display in the Museum of Natural History in Oxford is about the real Alice - the character in the books having been based on a real person.  Well, this post is about THE REAL LEWIS CARROLL.  His name was actually Charles Lutwidge Dodgson (1832-1898), and mathematics was his "day job."  He earned the Christ Church, Oxford, Mathematical Lectureship in 1855 and held it for the next 26 years.  His Fellowship at Christ Church, Oxford, gave him the privilege of living at Christ Church for the rest of his life provided he did not marry and that he prepared for holy orders, which is simply what was expected of Fellows living at the college at that time.

Museum of Natural History, Oxford

In 1856, Henry Liddell became dean at Christ Church.  Dodgson (Carroll) became very close to the Liddell family, especially Henry's wife Lorina and three of their children - the daughters Lorina, Edith and Alice.  He would sometimes take them to the Museum of Natural History (pictured above and below), and some of the characters in the Wonderland books came from creatures they saw here.  In particular it is likely that Dodgson represented himself as a Dodo because he had a stammer, so his name sometimes came out "Do-do-dodgson."

Museum of Natural History, Oxford

Dodo skeleton and taxidermy at Museum of Natural History, Oxford

Lewis Carroll enjoyed entertaining others.  This began early in life when he entertained his siblings.  He even used puzzles and riddles in his mathematics lectures at Oxford in order to engage his students.  He especially loved entertaining children.  I noticed in reading a biography of him that he used many of the same magic tricks that Mathemagician Arthur Benjamin, professor at Harvey Mudd College, uses today.

His friendship with the Liddell family involved excursions by boat, first with the son, Henry, and later with the daughters.  On July 4, 1862 Carroll invented the outline of the story we now know as Alice in Wonderland while out rowing with a friend and the girls.  Alice pressed him to write it down, and, after a long time, he finally presented her with a manuscript of the story, originally titled Alice's Adventures Under Ground.  

Writer George MacDonald, a pioneer of fantasy writing, was a friend and mentor of Lewis Carroll (and also influenced C. S. Lewis, J. R. R. Tolkien, Madaleine L'Engle and others).  It was after the family of George MacDonald had read the manuscript and after the MacDonald children had received it especially enthusiastically that Carroll took the manuscript to a publisher.  The rest, as they say, is history!

From the Alice display in the Museum of Natural History, Oxford

Another influence on Carroll was the pre-Raphaelite circle, including the Rossetti family - Dante Gabriel and Christina.  Below is an imagine of the Rossetti family plot in Highgate Cemetery, London.  Christina is buried here, and her name is on the flat stone on the ground.  The picture below is of a fox in the cemetery, which is a haven for wildlife.

Rossetti Family Plot - Highgate Cemetery, London

Fox in Highgate Cemetery, London (west)

Had Carroll not been remembered for his writing, he would have been remembered as the second-greatest photographer of the Victorian Era and the greatest photographer of children.  He was a very gifted artist, but, sadly, his reputation (as a person) has waxed and waned over the last century.  Many photographs he took of children were nude photographs, but he always took such photographs with the permission of the parents.  Also, it was the fashion of the time to take such photographs.  His writing too has gone through the scrutiny of Freudian psychology long after the fact, and people have come up with some pretty "interesting" and not so complimentary interpretations.  I side with Carroll biographer Robin Wilson who writes, "Sadly much nonsense has been written about Dodgson's friendships with children  .  .  .  .  Subjecting him to modern 'analysis' rather than judging him in the context of his time is bad history and bad psychology and says more about the writer than it does about Carroll."  

To that I say, "Amen!"

Here are some pictures of Christ Church College, Oxford.

Christ Church Cathedral as seen from the cloisters

Organ of Christ Church Cathedral

Court of Christ Church College, Oxford

Entrance to Christ Church, Oxford as seen from the road

A view of the entrance of Christ Church College from the court

In terms of mathematics, he worked particularly in geometry, logic, linear algebra and recreational mathematics.  Dodgson discovered a method for finding determinant of matrices that significantly shortened the process, especially for matrices of order 4 or larger; it is called his method of condensation.  He did excellent mathematical work with ideas relating to electoral reform.  In general, however, he was a very conservative mathematician who did not really break new ground, so none of his books have proven to be of enduring importance.

I am going to touch on two of his books, however.  I have enjoyed looking into The Game of Logic, which seems to me to include a method that would be an improvement on Venn Diagrams in working with syllogisms.  Another book that has endured has done so for historical purposes, and that is his work Euclid and His Modern Rivals.  Then, as now, there were arguments about how mathematics (and other subjects) should be taught.  For centuries Euclid's Elements was taught as if it were the Bible - with students actually memorizing its books and propositions by number as if they were chapters and verses in the Bible.  Others in his day wanted to use different geometry texts in which exploration and discovery were emphasized more than memorization, which sounds quite familiar to me in what is going on today with our pendulum swing between "back-to-basics" movements and "discovery-based" learning, as we are seeing right now with the Common Core fights raging around us.  Clearly this is nothing new.

I recommend Robin Wilson's 2009 biography Lewis Carroll in Numberland: His Fantastical, Mathematical, Logical Life.  I especially enjoyed excerpts from Carroll's (or should I say "Dodgson's") diary in which he expresses his thoughts about his teaching - his feelings when things went well and his self-doubt when things went poorly.  I found it impacting, eye-opening and touching to read about Lewis Carroll in Lewis Carroll's own words.

Nine Men's Morris Game

My family and I are gamers, and I love classic games like Chess, Go, Backgammon and Nine Men's Morris, so at the very beginning of my trip when I was visiting Jedburgh Abbey in Scotland and I saw an archaeological exhibition of a Nine Men's Morris game found at the site, I took a picture of it just for my own self  .  .  .

And then, I saw another Nine Men's Morris board at the home of Sir Isaac Newton in Woolsthorpe - not original, of course, but there it was in the living room!

The day after my arrival in London I went to a math lecture on Turing and von Neumann at the Museum of London, and I saw another Nine Men's Morris Board that had been excavated from early times in London.  I am shocked as I go through my pictures tonight to see I didn't take a photograph of it, but in it's place I'll post this picture from that museum of dice from centuries ago in London - just to illustrate how games have always played an important role in human culture.

Then yesterday I was at Hampton Court Palace, and in the Great Waiting Chamber from the Tudor Era, what did I see but another Nine Men's Morris game in the midst of this glorious splendour!

And then today I was in the Tower of London, and there was Nine Men's Morris again in the room set up as a replica of that of Kind Edward I.

There is no real relationship here to the math history I'm studying for my sabbatical and this game, BUT when parents of young children ask me (as happens so often!) what they can best do to help their kids be ready to be solid in mathematics, I always encourage them to play strategy games with their kids.  Such games are all about problem-solving and creative thinking, which is so necessary in mathematics - and by "mathematics" I don't mean merely arithmetic and number crunching - I mean solution and proof and Mathematics.

It's also the case that I choose a different strategy game every semester for my math tutor training sessions - a game that I ask the tutors to focus on all semester to see if they can find any mathematical principles relating to it that could ensure a win for them every time or at least an advantage. It's a way of getting them to stretch their "mathematical muscles" outside the classroom and more flexibly, in broader context.  Nine Men's Morris may need to be my next game pick for tutor training.

Anyway, Nine Men's Morris keeps popping up everywhere I look, so it gets a post.  Now I'm eager to pull out my Nine Men's Morris board when I get home so that I can play it again.  It's been a long time, but it's clearly a classic that I need to revisit!!

LATER ADDITION:

And yet another one!  This time it isn't in a museum but in a game store in Heidelberg, Germany.

And yet again another one!  This one was in my stop after Heidelberg - Gottingen, Germany!

AND EIGHT YEARS LATER (May 2024) 

I continue tracking down historic mathematicians and keeping an eye out for classic games as I do. While staying at a B&B in Venice I came across a book: 111 Places in Venice that You Must Not Miss. In it was a page on two Nine Men's Morris carvings to be found in the city, and so off I went to hunt for them.

Despite the information in the book, they were not easy to find, but the hunt was fun and worthwhile. The first one is carved into a bench just to the left of the entrance of Scuola Grande di San Rocco. The trouble with it being on a bench is that people sit on benches. Thankfully, the carving was peeking out to the side of someone's behind - yes, I was a little less observant than I usually am of personal space, and I just pointed to it and asked graciously if they would move so I could get a picture. (I will do almost anything in search of "games in the wild.")

It's just to the left of the lady with the pink purse. The guy sitting on the left with the phone had been sitting partly on the gameboard and kindly moved for me. It was clear from our conversation that someone else had asked him to move earlier, so, obviously, I am not the only "game hunter" out there.

This one at least had pretty specific directions in the book but for the second one, all I knew was that it was on the first floor (European "first floor" = American "second floor") of a shopping center: Fondaco dei Tedeschi. That's a pretty big area, so this one was quite a search, but I persevered and found it behind a display of clothing. I'm sure the store owner was amused (not). Either way, it made me very happy!

Pushing my way through this little display of clothes, I found:

It looks like this is two games in one. I'll update this when I recall the other game I have in mind. I want to say "Fox and Geese," but I don't think that's it:

Another update - in Madrid in April 2025, I saw another representation of Nine Men's Morris on a tote bag. I left my family behind, saying, "I have a picture I need to take; I'll be back." And I took off running after this woman. (I need a tote bag like this one!)

This game shows up everywhere - in ancient carvings, old (but not ancient) carvings, medieval game boards, modern board games, and modern tote bags. It is definitely a game with staying power.

Watch for further updates - with further travels may come further discoveries!

The Royal Society

When my students do presentations on mathematicians in courses that involve the history of mathematics, there are certain places that come up frequently.  It is important to me on this trip to visit those places that come up most often so that I can share my experience with my students to make these places seem more real - rather than just a name or something that may as well be on the moon!  One of the these places is the Royal Society of London, which I visited today.

Sometimes they give public lectures here at the Royal Society, and I'd been watching their site for a number of months now to see if I could take in such a lecture while here, but nothing came up.  At the very least I wanted to walk past the building and take pictures.  It is located just off The Mall - just down from Buckingham Palace - across from St. James' Park but facing away from the park.  The address is 6-9 Carlton House Terrace.

Behind the Royal Society - looking down The Mall toward Buckingham Palace

I almost just took pictures and kept going, but after having been to the Royal College of Physicians earlier in the week and having seen the exhibit on John Dee there, I decided to walk in and just ask if they had an exhibit or anything.  I introduced myself as being on sabbatical relating to mathematics and wondering if they had any sort of exhibit.  Sure enough, there is an exhibit on Micrographia, which isn't math, but was as good an excuse as any to get through the doors!

It was actually quite interesting, and it involves the work of Robert Hooke - he of the long-standing feud with Newton (which you can read a little bit more about in my post on Newton), so there's at least a tangential math connection.  Also displayed was a bound copy of Newton's letters to the Royal Society outlining the main results of his study of optics, particularly with regard to light and color.

Hooke's illustrations of crystals of frozen fluids - later included in his "Micrographia"

The sketches below are by Anton van Leeuwenhoek - considered the first microbiologist and the Father of Microbiology.  The second image is a close-up of his signature.

Now some Newton items - though I don't know why the computer won't let me flip the following image.  Anyway, it is his collection of letters to the Royal Society outlining his optical work, and the following photograph is of wood from his famous apple tree shaped into a prism.

The origins of the Royal Society are in the 1660s.  The first "learned society" meeting took place following a lecture at Gresham College by Christopher Wren.   The group soon received royal approval from King Charles II and from 1663 was known as "The Royal Society of London for Improving Natural Knowledge."

The Royal Society's motto is "Nullius in verba," or "take nobody's word for it."  It is an expression of determination to withstand the domination of authority and to verify all statements by an appeal to facts determined by experiment.  The first issue of "Philosophical Transactions" was published in 1665; this established the important concepts of scientific priority and peer review, and it is now the oldest continuously-published science journal in the world.  Among other things, the society published Newton's Principia Mathematica,  Benjamin Franklin's kite experiment, the first report in English of inoculation against disease, and they approved Babbage's Difference Engine.*

Sir Isaac Newton served as president of the Royal Society from 1703 to 1727.  Other presidents have included Sir Christopher Wren, Samuel Pepys, Sir Humphry Davy, Thomas Henry Huxley, The Lord Lister, and Sir Ernest Rutherford.

More recently the society awarded Peter Higgs the Copley Medal - its oldest and most prestigious award.

Do you recognize the pattern in the door handles?

The Royal Society arose out of informal meetings at Gresham College - the same Gresham College that has been giving free public lectures in London for more than 400 years, one of which I was able to attend earlier this week at the Museum of London to hear Gresham Professor of Geometry Raymond Flood present a wonderful lecture on mathematicians Alan Turing and John von Neumann.  (I couldn't believe how packed the hall was for this mid-day, mid-week math lecture!  I was lucky to find a seat!)

* Information from https://royalsociety.org/about-us/history/

John Dee

Crystal Ball of John Dee

 John Dee (1527-1609) lived during a time when science and magic were not well-distinguished.  In fact, during his time a scholar might not be considered to be such if he were not able to tell horoscopes or cast out devils.  As English commentator William Vaughn wrote in 1600, "Now-a-dayes among the common people, he is not adiudged any scholar at all, vnlesse hee can tell mens Horoscopes, cast out diuels, or hath some skill in southsaying."

On the other hand, in mid-1500s England, mathematics could be considered to be magic, and Dee was arrested in 1555 for "calculating."

Dee was closely associated with the court of Queen Elizabeth I.  He was an adviser to her and may have acted as a spy for her as well.  His mathematical work was used by navigators of the time.  He also made suggestions regarding calendar reform to Elizabeth, and they were excellent suggestions, better than what was proposed by Pope Gregory XIII, but the Archbishop of Canterbury was opposed, so these were not put into place. Dee was particularly known for his work in geometry.  He wrote a long preface to the first English edition (1570) of Euclid's Elements - a book that was issued with sheets of templates for three-dimensional diagrams to be cut out and pasted in (i.e. a pop-up book!).  He lectured in Paris on Euclid's Elements where people flocked to his lectures!

 As a student, mathematics was Dee's passion.  He studied 18 hours a day, allowing 4 hours for sleep and 2 hours for meals.  He took his degree at St. John's College, Cambridge, and he became a founding Fellow of Trinity College, Cambridge, which was founded by Henry VIII in 1546.

John Dee

In his diary, Dee claimed that the crystal pictured above was given to him by the angel Uriel in November 1582.  He used the crystal for predicting the future by looking for symbols or "ghosts" of people in the stone.  He, and his friend Edward Kelley, also claimed that Uriel instructed them about how to make the philosopher's stone.

The above may have been Dee's scrying mirror, something he may have used for conjuring visions, calling spirits, or divination.  Dee was so high in Elizabeth's favor - at least early-on - that she actually visited his home in Mortlake.  She is said to have viewed herself in his mirror and to have been pleased at what she saw.

Dee spent a great deal of time amassing a personal library.  He was abroad a great deal collecting books (and probably spying for Elizabeth at the same time).  However, during one of his travels, his home and library were ransacked, many or most of his books were taken, along with his instruments.  This exhibition, which is being put on by the Royal College of Physicians, is titled Scholar, Courtier, Magician: The Lost Library of John Dee.  I should have paid more attention to the details - no one knows who plundered Dee's library.  I believe some of his books ended up in the hands of one of his students eventually and then most of them found their way into someone else's library - a library that was eventually donated to the Royal College of Physicians who is now displaying it along with many personal items, such as the mirror and crystals pictured above.

Dee's life ended in poverty and obscurity.  He was never able to regain his fortunes after the plundering of his home, and he had fallen out of favor in court because of his increasingly deep explorations into the occult.  He was during his lifetime, and even more so since his death, a cult figure.  He is thought to have inspired Shakespeare's conjuror Prospero in The Tempest, Christopher Marlowe's Dr. Faustus, and Ben Johnson's The Alchemist.  Today there are works of art and literature - as well as a comic book character and a rock opera - based on him.  To fill in details of my sabbatical experience, I'm reading a recent novel based on his life - having to do with a fictional search for the bones of King Arthur in Glastonbury at the behest of Queen Elizabeth I.

An interesting character indeed!