From Gauss to Riemann: on the Shoulders of Euclid

This is a story of a teacher who considered teaching a “burdensome and ungratifying business” and a student who was shy and modest. While this was no trip like “Good Will Hunting”, they managed to take the world of mathematics by storm.

Carl Friedrich Gauss (1777-1855) was a child prodigy. You might have come across some of his childhood stories. One goes like this: On one fine Saturday, his father was calculating weekly payroll for a group of labourers. Carl was about 3. Instead of doing what other kids of his age are expected to do – “spilling a glass of milk and crying all over”, he pointed out a mistake in his father’s addition. His parents did not teach him arithmetic. In fact, nobody did. He came from a poor family and his mother was illiterate. His father was a hardworking and honest man, but he didn’t do anything to nurture Carl’s talent. He assigned him the weekly task of checking the payroll arithmetic and occasionally amused his friends with the freak toddler.

Another famous story dates back to the time when he was in his first school. Gauss himself described it as “squalid prison” or “hellhole”. His master, Buettner (German for “Do as I say or I’ll whip you”) presented students with tall calculations and kindly did not share his formulae with the class. One day he assigned the problem of finding the sum of numbers from 1 to 100. Carl was the only student to do it correctly, finishing an hour before the 49 others. Buettner was astounded and quickly appreciated him.

As a student in Göttingen (1795), he started taking interest in Euclid’s parallel postulate. There were quite a few failed attempts to prove it. However, Gauss was the first one to accept the fact that the postulate might not hold. To his teachers, it seemed that only a crazy person would doubt its validity. For the mathematically inclined, Euclid’s postulate goes something like this:

“Given a line segment that crosses two lines in a way that the sum of inner angles on the same side is less than two right angles, then the two lines will eventually meet (on that side of the line segment).”

Makes sense right? But as it turns out, our intuition can be quite deceiving. This postulate is valid only in a certain type of geometry known as “Euclidean space”. In non-Euclidean geometries such as Hyperbolic or Elliptic spaces, this postulate does not hold. The fall of the fifth postulate saw the genius of Gauss and the curved space revolution.

This was just the beginning. With the fall of the parallel postulate, the foundations of geometry and mathematics trembled. It was a point where what is true could no more be reckoned with day-to-day experiences. Amidst all this, Georg Friedrich Bernhard Riemann (1826 – 1866), still 19, matriculated at the University of Göttingen, where Gauss was a professor. Riemann was brilliant but shy. He changed his subject from theology to mathematics. His PhD dissertation earned him appreciation from, among many others, Gauss.

Though Gauss never whipped any student himself, he inherited the same appreciation for genius and scorny attitude from Buettner. Riemann secured a lectureship at the University. That job, unlike today, didn’t pay him any salary. To Riemann, it was still an honourable position, a stepping stone to professorship. And students gave tips. The only final test before him was a trial lecture. He had to submit three topics out of which it was customary for the faculty to choose the first one. Just in case, Riemann was prepared for both his first and second. Gauss, for his own amusement and delight, chose his third.

Riemann’s next step was something that everyone who has ever known or been a student would understand – he spent several weeks having a nervous breakdown, staring at the walls. For his third choice, he chose a topic he had some interest in, but very little knowledge. It was titled – “On the Hypotheses Which Lie at the Foundations of Geometry”. This topic was close to Gauss who was seriously ill by then. Fortunately, Riemann was able to pull himself together and presented his lecture on June 10, 1854. It is now considered as one of the great masterpieces of mathematics. He was too smart to be one of us after all.

His lecture was articulated in terms of differential geometry. As you might have guessed by now, curved spaces were difficult to reconcile with daily experiences because human beings see distances in terms of meters and kilometres in a universe which operates on levels of billions of light years. A small enough patch of such a space can be approximated to Euclidean space with sufficient accuracy. Riemann explained how a sphere could be interpreted as a two-dimensional elliptic space. There were problems with his model, which were addressed later. During his lifetime, his work had no great impact, although it became the most important mathematical tool for Einstein to frame his general theory of relativity. So was the transition from Euclid’s “common notions” to the new radical “curved space”.

 

Deepnika Jain

Third Year MME

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