## Wednesday, April 30, 2014

### Carl Friedrich Gauss: a birthday

Johann Carl Friedrich Gauß – one of the history's best mathematicians – was born on April 30th, 1777 (thanks, Moinak), to a poor working-class family in Northwestern Germany. If you are intrigued by the high number of "sevens" in his birthday, let me mention that he died when he was 77, too. Gauß on the ten-deutsche-mark banknote became an early victim of the euro (the banknote was worth 5.113 euros).

His contributions are huge: you may spend an hour just by enumerating things named after him – including things used by physicists every hour. The Gaussian distribution, Gauss-Bonnet theorem, the CGS unit of the magnetic field $$\vec B$$ (one gauss is $$0.0001$$ tesla where tesla is an SI unit), the 19th century CGS "Gauss" units, and tons of other things – not to mention dozens of theorems in mathematics – number theory, algebra, statistics, analysis, and differential geometry.

He would be married twice. The first wife died very early which made him rather bitter. Nevertheless, he has had 6 children. Wilhelmina had a chance to be a good mathematician but she died early. He insisted that the sons wouldn't become mathematicians because their subpar qualities would damage the brand "Gauß". Some of his job-related arguments with his sons forced one of the young boys to emigrate to America where he was successful.

Gauß didn't believe in a personal God or the Bible, he was a deist (God is fully seen by observing the natural phenomena) who promoted religious tolerance.

His basic school teacher, J.G. Büttner, wanted to get rid of the smart cookie named Gauß for an hour. So he told him to find the sum of integers between $$1$$ and $$100$$. Within seconds, Gauß told him it was 5050. The teacher was stunned. But you know, the average number is $$(1+100)/2=50.5$$ and there are $$100$$ of them, so the sum is $$100\times 50.5=5,050$$. Or if you want to avoid the average, combine the numbers to $$50$$ pairs like $$101=1+100=2+99=3+98=\dots$$ and these $$50$$ pairs sum up to $$50\times 101 = 5,050$$. Nice but not hard.

In fact, when he was around 20, he had already made his greatest contributions to number theory, the construction of polygons with a straightedge and a compass (a 17-gon on the grave was refused because it looks like a circle and is hard to create, anyway). He would prove the fundamental theorem of algebra (an $$n$$-th order polynomial has $$n$$ complex roots, some of which may coincide).

In his middle years, 1799-1839, he would play with the curves and co-discover non-Euclidean geometries although he didn't publish that result (but his extensive discussions of parallel lines in his letters make it rather clear that he had found everything that was needed to establish the field of non-Euclidean geometry). His "brother under the banner of the truth" Farkas Wolfgang Bolyai found it as well and did publish it, however. The priority dispute has weakened their friendship.

It doesn't mean that Gauß did all important things when he was young enough. He became a physicist only when he was older. In fact, when he was 54, in 1831, he would begin his co-operation with Wilhelm Weber. He would understand the number of independent electromagnetic units (you don't need any new units at all if you set $$\epsilon_0=1$$ or something like that) and he coined his own (Gauss') CGS system(s) of units in electromagnetism. He would also find the Gauss' law, i.e. Gauss' flux theorem – the net electric flux through a closed surface is equal to the electric charge inside the surface. This theorem also gives a relationship between the differential and integral forms of the equation $${\rm div}\,\vec D = \rho$$ that would later become one of the simpler Maxwell's equations.

(The unit "one gauss" in CGS was superseded by "one tesla" in the SI units. The latter is equal to $$10,000$$ gauss. The SI unit of the magnetic flux of 1 weber is equal to the "tesla times squared meter".)

In fact, his "physical" contributions were diverse. In 1818, he would perform a geodesic survey of the Kingdom of Hanover where he has lived most of his life. He has used his own invention, the heliotrope (a mirror used to reflect light and measure large distances).

He has also made important elementary contributions to optics – well, geometric optics or more precisely Gaussian optics, the set of laws how the rays change their directions if they enter lens under angles that are small, $$\vartheta\ll 1$$. When he was 23, he would help to pinpoint the newly discovered dwarf planet Ceres – by some tricks involving conic sections, their intersections, and timing. When he established himself as an astronomer, he didn't hesitate to accept the job of the Professor of Astronomy and the Director of the Göttingen Astronomical Observatory.

Gauß is also a key person in statistics. He has said that he had been using his "method of least squares" – and that's very important – since 1795. He has also realized the simplifications that occur if the errors are normally distributed – i.e. if they follow the Gaussian distribution, $$\rho\sim \exp(-x^2/2\Delta x^2)$$. Adrien-Marie Legendre published these results in 1805 so that Frenchmen became the "publication-official" inventor of these things.

He has been a very important man and I think it's too ambitious a project to write a meaningful yet original enough biography so let me give it up and stop.