## Friday, January 2, 2009

### Gauss

Karl Friedrich Gauss was a German mathematician whose career spanned the 18th and 19th centuries. He was well known in his time as the "Prince of Mathematics" and as the "greatest mathematician since antiquity," but he's largely unknown today (except by mathematicians, both professional and amateur).

Maybe the most well-known anecdote about Gauss is one that happened when he was just a boy. When Gauss was in school at age 10, his teacher, perhaps needing a quiet half hour, gave his class the problem of adding all the integers from 1 to 100. Gauss immediately wrote down the answer on his slate (this would have been in the year 1787). He had quickly noticed that the sum (1 + 2 + 3 + ... + 100) could be rearranged to form the pairs (1 + 100) + (2 + 99) + (3 + 98) + ... + (50 + 51), and that there were 50 pairs each equaling 101. This reduced the problem to the simple product 50 x 101 = 5050.

The fact that Gauss isn't as well-known as scientific giants, such as Archimedes, Newton, Einstein, and Hawking, is curious, since his accomplishments equal (some might say surpass) even those great scientific minds. Consider the following accomplishments:
• In 1796, at the age of 19, Gauss proved the constructability of a heptadecagon (a regular polygon with 17 sides) using only a straghtedge and a compass. Greek philosophers had believed this construction was possible, but a proof escaped them. Note that Guass proved the construction was possible without actually providing the steps necessary. The first explicit construction of a heptadecagon wasn't given until 1800, by Johannes Erchinger. Gauss was so proud of the proof of the construction that he requested the shape be placed on his tombstone. The stonemason refused, stating that the shape would have been indistinguishable from a circle.
• In 1799, in his doctoral dissertation, Gauss provided a proof for what is now known as the Fundamental Theorem of Algebra, which states that every polynomial has at least one root that is a complex number. Gauss would go on to provide four separate proofs of this theorem throughout his career.
• In 1818, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances. Gauss used the device to aid him in the land survey of the Kingdom of Hanover.
• In 1833, together with Wilhelm Weber, Gauss built the first electromagnetic telegraph used for regular communication, in Göttingen, Germany. This accomplishment is often attributed (by many Americans) to Samuel Morse, who independently developed and patented an electrical telegraph in the U.S. in 1837.
• Gauss lived and worked by the personal motto "pauca sed matura," or "few but ripe." Consequently, he published only a small fraction of his work. Years after his death, publication of his diary and letters confirmed that Gauss was the earliest pioneer of non-Euclidean Geometry, which is normally (and rightly) attributed to Bolyai and Lobachevsky (the two men developed non-Euclidean geometry independently from Gauss, and one another).
Taken individually, any of these accomplishments would have secured Gauss a spot in history, some of them just a footnote, but many of them a full chapter. Take them together, and you can see why Gauss is considered by many to be one of the true giants of scientific thought.

Jeff Moser said...

Gauss did a lot of great things for algebra and statistics. I first really found out about him as a kid reading E.T. Bell's "Men of Mathematics."

I'm curious though: if you had the choice of understanding everything Gauss did or everything Euler did, which would you pick?

Bill the Lizard said...

Jeff,
That's an excellent question. I'm going to have to defer answering until I've had a chance to do the same kind of research on Euler as I've recently done on Gauss. Thanks for giving me the subject of my next mathematician profile article. :)

Jeff Moser said...

I'll look forward to answer ;)

Bill the Lizard said...

Jeff,
I finally finished my (very short, considering all of his work) article on Euler.

You asked, "if you had the choice of understanding everything Gauss did or everything Euler did, which would you pick?"

After reading about both men and their work, I'm going to have to go with Gauss. His life and career were later, and in a lot of cases he extended work that Euler started. In understanding everything that Gauss did, you would understand the majority of what Euler did as well.

I wish I had two lifetimes to spend learning all of the math that both of them gave us. I suspect it would take me as long.

Thanks again, for posing such a fantastically thought-provoking question.

Anonymous said...

I am doing a research on Gauss' work and would like to thank you for posting such a nice summary of his accomplishments (short, but eloquent).
Regards,
Fabio (from Brazil)

Anonymous said...

So what's the complex root of the polynomial P(x) = 1?