You Can Count on It... A Brief History of Tally Sticks

The term "tally" comes from the name of a stick on which counts were made to keep a count or a score. The Latin root is talea and is closely related to the origin of tailor, "one who cuts". Many math words have origins that reflect back to the earliest and most primitive uses of number. Compare the origins of compute, digit, and score. The first record existing of tally marks is on a leg bone of a baboon dating prior to 30,000 BC. The bone has 29 clear notches in a row. It was discovered in a cave in Southern Africa. It is sometimes called the Lebombo Bone after the Lebombo mountains in which it was found. The exact age of such artifacts is a subject of debate, and their mathematical usage is somewhat speculative. Some sources have stated that the bone is a lunar phase counter, and by implication that African women were the first mathematicians since keeping track of menstrual cycles requires a lunar calendar.
Another candidate for the oldest tally record in history is a wolf bone found in Czechoslovakia with 57 deep notches cut into it, some of which appear to be grouped into sets of five.
In Mathematics Galore by Budd and Sangwin, there is a story of much more recent tally sticks. It seems that until around 1828 the British kept tax and other records on wooden tally sticks. When the system was

discontinued they were left with a huge residue of wooden tally sticks, so in 1834 they decided to have a bonfire to get rid of them. The bonfire was such a success that it burned the parliament buildings to the ground. What Guy Fawkes could not do with dynamite the Exchequer did with tally sticks.... The power of math.
The story, as improbable as it seems, is verified by a speech by Charles Dickens 1855. [Charles Dickens, Speech to the Administrative Reform Association, June 27, 1855, in Speeches of Charles Dickens, ed. K.F. Fielding, Oxford: The Clarendon Press, 1960, p. 206, ] The somewhat clipped version below is taken from Number, The Language of Science by Tobias Dantzig (pgs 23&24)

Ages ago a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer and the accounts were kept much as Robinson Crusoe kept his calendar on the desert island. A multitude of accountants, bookkeepers, and actuaries were born and died... Still official routine inclined to those notched sticks as if they were pillars of the Constitution, and still the Exchequer accounts continued to be kept on certain splints of elm-wood called tallies. In the reign of George III an inquiry was made by some revolutionary spirit whether, pens, ink and paper, slates and pencils being in existence, this obstinate adherence to an obsolute custom ought to be continued, ..... All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took until 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur ot any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However, they never had been useful, and official routine required that they should never be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the paneling; the paneling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost therof.

Several images of the fire was painted by J.M.W. Turner who watched the fire from a boat on the Thames. I have a clip that I can not credit that says, "The fire of 1834 burned down most of the Palace of Westminster. The only part still remaining from 1097 is Westminster Hall. The buildings replacing the destroyed elements include Big Ben's tower (oooh, side bar... Big Ben is not the name of the tower at Westminster, it is the name of the great Bell in the Chimes there.. admit it, you did NOT know that), with it's four 23 feet clock faces, built in a rich late gothic style that now form the Houses of Commons and the House of Lords. These magnificent buildings are still the subject of many paintings, including my own Parliament, with the grand Westminster Abbey on their north." The one below hangs in the Tate Gallery; while another, I believe, is in a gallery in Cleveland, Ohio.

*WIkipedia

Around 1960 an ancient mathematical record on bone was uncovered in the African area of Ishango, near Lake Edward. While it was at first considered an ancient (9000 BC) tally stick, many now think it represents the oldest table of prime numbers. Here is a link with a picture where you can see and read more about the "Ishango bone"

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Typing Monkeys

More observations stimulated by John Barrows new book (see my recent blog) Everybody has heard the suggestion that a million (or some other number) of monkeys typing continuously for many millenia would eventually produce a) Shakespeare, b) all of known science, c) the bible, d) all of the above). It began with Jonathon Swift and Gulliver's Travels, 1872, according to Professor Barrow. In the tale "a mythical professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters..." (I think, see emphasis in the excerpt below, that it was random strings of words).. Anyway, according to the good Professor Barrow, the story was embellished in different forms until French Mathematician Emile Borel{there is a street and a square named for him in the 17th District in Paris} suggested that random typing monkeys could duplicate the French national library. A few years later(1929), Arthur Eddington Anglicised that to "books in the British Museum." By 1972, Arthur Koestler writing in The Case of the Midwife Toad, New York, 1972, page 30, refers to Monkeys typing Shakespeare as "proverbial":"Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits--to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet." Ok, so eventually someone had to put this to a more scientific test, and they did. "A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:"RUMOUR. Open your ears; 9r5j5&?OWTY Z0d... " Even more impressive, to me, is the fact that 'in another part of that book, Swift tells of how the astronomers on the flying island of Laputia had: "discovered two lesser stars, or satellites, which revolve around Mars, whereof the innermost is distant from the center of the primary exactly three of his diameters, and the outermost five: the former revolves in the space of ten hours, and the latter in twenty-one and a half". Swift wrote this in 1726, but it was not until 1877 that Asaph Hall discovered the two moons of Mars.'... I just did a little checking on the orbit and periods he "predicted?" and the actual periods are about 7 hours for Phobos, and about 30 for Deimos... and their distance from the planet were about 9 x 10³km and 23.5 x 10³km. Mars has a diameter of 6.794 x 10³km so they are closer to 1.5 and 3 radii away it seems, but wow, for 100 years before the actual discovery??? Don't you wonder what made him use Mars instead of Venus or ??? Wait, maybe authors typing randomly can describe the true nature of the universe (with some limits of error)... ------------------------------------------------------------------------------------------ Ok best guess,  Swift had heard of a misinterpretation of Johannes Kepler. He had interpreted the anagram that Galileo Galilei sent to him in 1609 to tell him about the discovery of the phases of Venus as the discovery of two moons of Mars. 

 If you have your copy, here is what I found in Chapter Five of Gulliver's Travels. The whole thing is available at the Guttenburg Project. "The first professor I saw, was in a very large room, with forty pupils about him. After salutation, observing me to look earnestly upon a frame, which took up the greatest part of both the length and breadth of the room, he said, "Perhaps I might wonder to see him employed in a project for improving speculative knowledge, by practical and mechanical operations. But the world would soon be sensible of its usefulness; and he flattered himself, that a more noble, exalted thought never sprang in any other man's head. Every one knew how laborious the usual method is of attaining to arts and sciences; whereas, by his contrivance, the most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." He then led me to the frame, about the sides, whereof all his pupils stood in ranks. It was twenty feet square, placed in the middle of the room. The superfices was composed of several bits of wood, about the bigness of a die, but some larger than others. They were all linked together by slender wires. These bits of wood were covered, on every square, with paper pasted on them; and on these papers were written all the words of their language, in their several moods, tenses, and declensions; but without any order. The professor then desired me "to observe; for he was going to set his engine at work." The pupils, at his command, took each of them hold of an iron handle, whereof there were forty fixed round the edges of the frame; and giving them a sudden turn, the whole disposition of the words was entirely changed. He then commanded six-and-thirty of the lads, to read the several lines softly, as they appeared upon the frame; and where they found three or four words together that might make part of a sentence, they dictated to the four remaining boys, who were scribes. This work was repeated three or four times, and at every turn, the engine was so contrived, that the words shifted into new places, as the square bits of wood moved upside down. Six hours a day the young students were employed in this labour; and the professor showed me several volumes in large folio, already collected, of broken sentences, which he intended to piece together, and out of those rich materials, to give the world a complete body of all arts and sciences; which, however, might be still improved, and much expedited, if the public would raise a fund for making and employing five hundred such frames in Lagado, and oblige the managers to contribute in common their several collections. He assured me "that this invention had employed all his thoughts from his youth; that he had emptied the whole vocabulary into his frame, and made the strictest computation of the general proportion there is in books between the numbers of particles, nouns, and verbs, and other parts of speech."

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.
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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).
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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...

The Tale of the Creole Pig

This is not a math blog, except that it has to do with logic, or the lack thereof, but I teach kids, and this is a story my bright kids need to read.....and thanks to JD2718 in New York for passing this along.

Just before the recent storm hit Haiti, Kendra Pierre-Louis wrote this blog about the Creole Pigs that were once literally everywhere in Haiti, and how they came NOT to be there. It is a story of the worst indifference to sustainable development, and needs to be shared. ... This is me sharing. Since Blogs come and go, I am copying the whole thing below, but I do encourage you to read the original:
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Growing up in the United States, I grew up listening to my Haitian father speak longingly of two things that he said we couldn't get quite right in the US. The first were mangoes. Most of the mangoes that are found in the US are vaguely round like a Nerf football, and have a mostly deep reddish hue when ripe. They are beautiful, but to hear my father speak, are to the mango what the Red Delicious is to the apple: overproduced and vaguely generic.

The mango of his childhood, the Madame Francis mango, is flatter and green - like an overgrown lima bean. Even at its ripest it only hints at a dusky yellow color. It is also unique to Haiti. I've had it, and he's right; it is delicious; a queen among mangoes.

My father's other long lost food craving, pork from the Creole Pig, was also unique to Haiti. Unlike the pink pig encapsulated in the image of Wilbur, the pig from Charlotte's web, the Creole Pig was not pink. It, like the population of Haiti, was black and thus unlike American pigs did not sunburn. Raised by eighty to 85% of rural households, the relatively small but dense Creole pig subsisted not on grain, but on the detritus of the island's human population. It could thrive on the husk of rice, the cob of corn. In a nation without consolidated trash pickup the Creole pig acted as the nation's garbage men playing a key role in maintaining the fertility of the soil. And, because it was not dependent on feed for its survival, it functioned for the peasant population as a sort of mobile, literal piggy bank - the animals were sold or slaughtered to pay for school, for marriages, for unexpected medical expenses.

All of this is spoken in the past tense because between the 1970s and the 1980s the Creole pigs were systematically eradicated under pressure of the US government.

Like most of development history some of the facts are in contention, but this much is certain. In the 1970's the African Swine River Virus had spread from Spain to the Dominican Republic and then to Haiti by virtue of the Artibonite River which straddles the two countries.

Now comes the contentious part.

By 1982, says the United States government almost 1/3rd of Haiti's pig population was infected. A lot of Haitians (and many independent organizations) argue otherwise. What is not in contention is that the US, in fear of the virus spreading to its own pig population, pressured Haiti's government to seize all of the pigs and kill them.

Everyone who had pigs seized were supposed to be compensated in the form of replacement pigs - fat, pink pigs from the American Midwest, deemed 'better' by the USDA. These pigs needed clean drinking water (which 80% of Haitians did not have access to), 90 dollars a year in feed (in a nation where per capita income was 130 dollars a year), vaccination, and special roofed pens to serve as protection from the harsh Caribbean sun.

Does anyone see a problem with this?

Never mind the fact that many Haitians who had their pigs seized were never actually compensated (more on that in a second) - they couldn't have afforded the compensation anyway. In fact, many of those who received pigs found that their new pigs rapidly died.

So much for 'better'.

The eradication of the Creole pig only served to further impoverish Haitians. It forced many children to quit school, forced small farmers to mortgage and eventually lose their land, and forced many Haitians to cut down trees, rapidly increasing the Island's rate of deforestation, to create cash income from charcoal. All simply to save an already rich country from the small risk (and by most independent accounts the number of pigs infected in Haiti was much smaller than the 33% cited by the US) posed to it by a poor, tiny island nation.

It was, however, a boon to US pig farmers who generated millions in revenue according to grassroots international offloading these ill suited pigs on poor Haitian peasants. How?

In order to get a replacement pig, Haitians were required to pay a princely sum of $50 dollars per pig.