A Mathematical History Tour

(One of my favorite series in Scientific American  was James Burke's "Connections" (Later collected into the book Circles). This is a sort of homage to those, for math enthusiasts. There will be more.)

*Note: An earlier version stated that Archimedes was from Seneca, which was in error. Archimedes is from Syracuse. And I live in Ithaca, New York, which is south of Syracuse and east of Seneca Lake (both in New York). Hence my confusion! Mea culpa. (and thanks to Chris Maslanka for pointing it out.)

In 1906, Danish historian J.L Heiberg was inspecting a religious text from the thirteenth century, when he made an astonishing discovery – the traces of mathematical symbols that could be seen on the parchment were from the lost works of Archimedes of Syracuse. Archimedes was a third-century Greek mathematician who was known for his work in mechanics, and for the legend of his death (one account says that he was working on a mathematical problem during the siege of Seneca when a Roman soldier burst in to his study – Archimedes yelled out “don’t disturb my diagrams!” and the soldier responded by running him through with his sword. Seems a tad aggressive, if you ask me. Maybe the soldier had math anxiety.) In the thirteenth century, a manuscript containing transcriptions of Archimedes' work was scraped and washed to make a Christian liturgical text. (This was a pretty common practice in the early middle ages, since parchment was so expensive - these recycled texts are called palimpsests.) Often, traces of the original text would still be visible, and historians could restore the original text - so, nearly 700 years later, the Archimedes palimpsest was recognized for what it was, and the underlying text was translated. 

Among other works, the palimpsest contains a manuscript called “On the Sphere and the Cylinder.” In it, Archimedes gives a method for finding the surface area of a sphere which uses a technique that closely resembles a modern technique (Riemann summation) named after nineteenth century mathematician Bernhard Riemann – famous for the Riemann Hypothesis, one of six (originally seven) unsolved problems in mathematics for which the Clay Mathematics Institute offers a $1,000,000 prize for a proof or solution. You would think, given his inclusion on such a list, that he was one of those precocious whiz children that wowed their teachers with mathematical insights from the time he was a child, and (at least in this case) you would be right. However, Riemann initially went to university to study theology, and had to be convinced to drop it for mathematics by mathematical heavyweight Carl Friedrich Gauss. 

This is itself impressive, as Gauss, though an extremely prolific and influential mathematician, reportedly hated teaching and thus took on very few students in his life. Gauss was a perfectionist of the highest order, often failing to publish important mathematical results until after he died they were perfectly polished. As a result, mathematicians with exciting new discoveries in the field were often disappointed to find that Gauss had already proved their results (though I do wonder whether some of that was just talk on Gauss’s part – he tended to exaggerate his claims of priority in his correspondence). In one such case – that of Russian mathematician Nicolai Lobachevsky, and his work in non-euclidean geometry – Gauss claimed to have known of the results for 54 years! He even goes so far as to write in correspondence that “there was nothing materially new for me in Lobachevsky's paper.” (I raise my eyebrow at that claim, since Gauss reportedly collected all of Lobachevsky’s known work, and then taught himself Russian in order to read it. You don’t do all of that for “nothing materially new.”)

Lobachevsky’s name was also (perhaps unfortunately) immortalized by Tom Lehrer, a satirical singer-songwriter active in the 1950s and 60s. (Lehrer holds a special place in the hearts and minds of scientists everywhere for the song "The Elements", where he set the names of the chemical elements to the tune of the "Major-General's Song" from Gilbert and Sullivan's Pirates of Penzance.) Trained as a mathematician, Lehrer wrote many songs with mathematical themes, and “Lobachevsky” was one of these. It tells the story of a mathematician who climbed his way to the top of his field by plagiarizing the work of others, and then publishing before they could claim credit – this unnamed mathematician claims to have learned this skill from Lobachevsky (even though there is no evidence that Lobachevsky ever did such a thing):

Let no one else's work evade your eyes
Remember why the good Lord made your eyes
So don't shade your eyes
But plagiarize, plagiarize, plagiarize!
(Only be sure always to call it, please, "research.")

And ever since I meet this man
My life is not the same
And Nicolai Ivanovich Lobachevsky is his name! 

Lehrer is, as of this writing, still around, though he hasn’t written new music in decades. I would say that he’s 86 years old, but he prefers to calculate his age in Centigrade (so he’d say he’s 30, despite being born in 1928). The Centigrade system was developed (as a measure of temperature, not age!) by Swedish astronomer Anders Celsius, who travelled with Pierre Louis Maupertuis on an expedition to Lapland to determine the exact shape of the earth. The success of this expedition led to Maupertuis being tapped to run the Prussian Royal Academy of Sciences alongside a partially blind Swiss mathematician named Leonhard Euler. Euler, like Gauss, was an intensely prolific and formidable mathematician – his work covered (among other things) calculus, optics, fluid dynamics, astronomy, and even music theory. Euler tackled (and proved) Fermat’s Little Theorem, which Pierre de Fermat had stated without proving it himself (this was a bad habit of Fermat’s – one other theorem of his, found in the margins of a book after his death, took mathematicians over 350 years to prove). 

Just before he died, Fermat published a paper on analytic curves, which included a type of logarithmic spiral that became known as the Fermat spiral. Fermat spirals can be found as patterns in the heads of sunflowers, and is closely related to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.( For a long time, it had been noticed that these numbers were important in nature, but only relatively recently has it been understood that this is due to efficiency during the growth process of plants.)

This sequence was also studied extensively by French Mathematician Edouard Lucas, who, in his spare time, developed mathematical puzzles. The most famous of these was called the Tower of Hanoi – which, according to a paper published in 2010, can be solved by a species of Argentine ants using non-linear dynamics.  (Yes, ants. Doing non-linear dynamics. I find that to be a humbling thought, since I can barely keep up with my laundry.) Unfortunately, in 1891 Lucas attended a banquet that would spell his doom - while he was there, a waiter dropped a dining plate and one of the pieces bounced up and cut Lucas’s cheek. This being Europe in the late 1800s, Lucas died within the week due to a form of septicemia*. (Bummer.) 

Lucas was well regarded, and his obituary was printed in Popular Science Monthly the following January. This same issue featured a delightfully enthusiastic article entitled “The Aviator Flying-Machine” by one Gustave Trouvé, who was heralding the end of balloon flight – “let us always keep in mind that we shall thank it as soon as possible for its services and show it the door”  – and the start of a new era of heavier-than-air flying machines, like… mechanical birds. Don’t laugh – apparently, the study and design of mechanical birds was an important part of aeronautics research at one time. So much so, in fact, that aviation pioneer (and mentor to the Wright brothers) Octave Chanute not only reproduced Trouvé’s design for a mechanical bird in his book Progress in Flying Machines, but devoted a considerable section of the book to calculating speed and horsepower of various birds: 

"Napier assumed that a swallow weighing 0.58 oz. must beat his wings 2,100 times a minute while going 33 ½ miles per hour, in order to progress and sustain his weight, and that it therefore expended 1/13 of a horse power. In point of fact, the bird only beats about 360 times a minute, and is chiefly sustained by the vertical component of the air pressure on the under side of the wings and body… "

(Note: I have read the book cover to cover, and was disappointed to find that there was no mention of African vs. European, or laden vs. unladen swallows. Surely a great scientific opportunity was missed.) 

In this same book, Chanute describes the work of Sir George Cayley, an English aviation pioneer. This Cayley was the distant cousin of Arthur Cayley, a professor credited with founding the modern British school of pure mathematics. One of the many abstract concepts Arthur Cayley worked with were the quaternions, an algebraic group first described by Irish mathematician William Rowan Hamilton in 1843. Or, at least, the description was first published by Hamilton – a review of Gauss’s unpublished work shows that he worked on something similar as early as 1819. (I’m assuming that if he had been paying attention, he would have undoubtedly told Hamilton that he knew about the math behind the quaternions for 50 years or so, and there was nothing materially new in Hamilton's work. Remember Lobachevsky? Gauss was kind a of jerk like that.)

In fact, one of the precious few to receive unqualified praise from Gauss was an obscure French mathematician named Monsieur LeBlanc – in a letter to a colleague, Gauss writes that LeBlanc had studied Gauss’s works in number theory “with a true passion… and has sent me occasional very respectable communications about them.”  Gauss corresponded with Monsieur LeBlanc for three years before discovering that LeBlanc was actually a woman named Sophie Germain – upon discovering this, he wrote a letter to Germain, in which he expresses his admiration: 

"When a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny problems, succeeds nevertheless… then without doubt she must have the noblest courage, quite extraordinary talents, and a superior genius."

Germain did indeed have a great deal of courage and determination – she began to study mathematics as a young woman, confined to her family’s house during the French Revolution. Her family did not approve – going so far as to put out her fire (and even take away her clothing!) in order to force her sleep instead of study. She thwarted them by wrapping herself in blankets, and, waiting until everyone had gone to bed, studying by candlelight. What inspired her to pursue mathematics with such passion? She had read an account of the early death of a Greek mathematician who was killed by a Roman soldier – none other than Archimedes of Syracuse.

*According to his obituary, Lucas died of erysipelas, which was also commonly known as "Saint Anthony's Fire." Thanks to @OnThisDayinMath for this bit of info.

References (for anyone who wants them)

Chanute, Octave. Progress in Flying Machines. Toronto, CA: General Publishing Company,
Ltd., 1997.

Edwards, Harold M. Fermat's Last Theorem: A Genetic Introduction to Algebraic Number
. New York, NY: Springer Verlag, 1977.

Fellmann, Emil A., Gautschi, E., and Gautschi, Walter. Leonhard Euler. Basel, Switzerland:
Burkhauser Verlag, 2007.

Knoebel, Art. Mathematical Masterpieces: Further Chronicles by the Explorers. New York, NY:
Springer Verlag, 2007.

Osen, Lynn M. Women in Mathematics. Boston, MA: Massachusetts Institute of Technology,

Pickover, Clifford. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them.
New York, NY: Oxford University Press, 2008.

Reid, Chris R., Sumpter, David J.T., and Beekman, Madeleine. “Optimisation in a natural
system: Argentine ants solve the Towers of Hanoi.” The Journal of Experimental 
. January 1, 2011.

Stillwell, John. Mathematics and its History. New York, NY: Springer Verlag, 2010. 

Trouvé, Gustave. “The Aviator Flying-Machine.” Popular Science Monthly vol. 40. January