Carl Friedrich Gauss

Gottingen University Observatory (front)

Carl Friedrich Gauss (1777-1855) is known as the Prince of Mathematics and is generally considered to be one of the "top three" mathematicians of all time, along with Archimedes and Sir Isaac Newton. Gauss was the first director of the observatory in Gottingen, pictured above and below.  From 1807 until his death in 1855, Gauss lived in the rooms at the observatory - rooms later occupied by Dirichlet and then Riemann.

Gottingen University Observatory (back)

His aptitude for mathematics was evident early-on.  When he was 3 years old he discovered an error in his father's payroll calculations.  When he was 7 years old, his teacher, wanting to keep the students busy for a while, assigned the class to add the numbers 1 to 100.  Instead of doing this in the way most people would and in the way his teacher certainly must have expected, beginning with 1+2=3 and then adding 3 onto that result and then 4 and so on, Gauss somehow saw a faster, easier way of doing this and wrote down the correct answer 5050 almost immediately.  (Later in the post I'll explain how he did it, but I'm saving it for later in case you want to take a moment to see if you can find a short cut too!)

Gauss contributed to many branches of pure and applied mathematics and mathematical physics.  We, who love our phones, can thank Gauss for work he did very early on telegraphy.  Through the work of Gauss and his Gottingen colleague, physicist Wilhelm Weber, the first electromagnetic telecommunication was sent in 1833, predating Samuel Morse's telegraph by 4 years.  Below are photographs of memorials of this event on the grounds of the observatory - a monument and a plaque - as well as a statue elsewhere in town. These photos are followed by images relating to Weber - a street in Gottingen named for him, and his tombstone.

Wilhelm Weber tombstone - Stadtfriedhof Cemetery, Gottingen, Germany

Though Gauss made contributions to almost all branches of mathematics, his favorite seems to have been number theory.  He called mathematics the "queen of the sciences," but he called number theory the "queen of mathematics."  In 1798, at the age of 21, he wrote a textbook titled Disquisitiones Arithmeticae, which transformed number theory from a collection of isolated theorems and conjectures into a well-structured branch of mathematics.

He had a passion for prime numbers and would at times spend "an idle quarter of an hour" factoring a thousand consecutive numbers (a "chiliad" of numbers) in order to determine which were prime.  At the age of 19 he formulated his prime number theorem, which has to do with the distribution of prime numbers among the counting numbers.

In that same year he constructed a regular 17-sided polygon using a straight-edge and compass.  His proof that this could be done was the first progress in 2000 years in the area of constructing regular polygons.  The images below - window art and wall posters - celebrate this, but they are not from Gottingen, as the rest of the pictures in this post are, but rather from the Mathematical Sciences Research Institute in Berkeley, California.

Not only does MSRI have a display about Gauss's construction of the 17-gon, but it is the case that they are located at 17 Gauss Way!  How's that for homage to a great mathematician?

While Gauss worked in many areas of mathematics, he did not always publish his findings.  His motto was "Pacua sed matura" ("Few but ripe").  One example of this is his work in non-Euclidean geometry, and, unfortunately the tale has a bit of a sad ending.

A young man named Janos Bolyai, a Hungarian, also entered into the (at the time) strange and very new universe of non-Euclidean geometry, a geometry of curved space, which we now know to be the sort of geometry Einstein needed in order to formulate his theories of relativity.

Janos's father, Farkas, who was also a mathematician, warned him not to pursue this strange, new area - calling it an "endless night" that would consume all his leisure, his health, his peace and his joy in life.  But Janos couldn't leave this intriguing mathematics alone, and he ended up developing what we now call hyperbolic geometry.

Farkas, who had been a student of Gauss, wrote a mathematics textbook and included his son's work as an appendix; he then sent this to Gauss, his former teacher and the greatest mathematician of the age.

Gauss responded that he could not praise the work of Janos because, as he wrote, to "praise it would be to praise myself, for the entire content of the work, the methods that your son used, and the results to which he was led coincide almost completely with my own meditations from 30 to 35 years ago  .  .  .  My intent was not to let any of my own work on this, of which till now very little has been put on paper, be known during my lifetime  .  .  .  I am therefore surprised to learn that I have been spared the effort, and it is very pleasing to me that it is the son of my old friend who has anticipated me in such a surprising manner."  (This was no false boast on Gauss's part.  He had, in fact, made these discoveries years before and never published them.)

Upon receiving this response Janos was greatly upset to the point of physical and mental illness.  This promising young man, whom Gauss called a "genius of the first order" in a letter to someone else but not to Janos himself, gave up mathematics and remained impacted throughout life by what he felt to be a severe blow.  Gauss hadn't meant to discourage him, nor to belittle him; Gauss was just stating facts, but Gauss was rather lacking in tact in how he did so.

I could, and probably should, balance the scales by telling the story of Gauss's encouraging mentoring of Sophie Germain, but I have written a bit about it in my post about her life, which you can find by clicking on her name and then scrolling about three-fourths of the way down that post.  Yet though he mentored her kindly, at one point he just stopped responding to her letters when other things took his attention.  And then there's the story of his comment when he was in the middle of a math problem and was told his wife was dying  .  .  .  so it seems that tact was truly not a strong point for him.

The tower in the three pictures above, seen from Plesse Castle near Gottingen, is known as Gauss's Tower.  Though the current tower dates from 1964 it is near a spot used by Gauss in his work surveying Hanover, as commissioned by King George IV of England.  While doing this work, Gauss invented the heliotrope, an instrument that reflects sunlight across great distances.  (At this link is a nice essay about his work as a surveyor.)

I've mentioned that Gauss was the greatest mathematician of his age.  He first came to wide-spread fame in 1801 when he was 24 years old when his mathematical ability allowed him to predict where the newly discovered dwarf planet Ceres would reappear after disappearing, in its orbit, behind the glare of the sun.  Astronomers were thrilled at being given the ability to find Ceres again, because mathematical tools that they had access to at the time were not sufficient for them to extrapolate its future position given the small amount of data (1% of the orbit).

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SPOILER ALERT: I guess it's time that I should share GAUSS'S SHORTCUT for adding the numbers 1 to 100, as promised earlier.  Though most people think math is about making life hard; math is actually about making things easier, and this can often be done by finding a different viewpoint.  Instead of plowing through and adding the numbers in order, Gauss realized that if he added bigger numbers and smaller numbers it would simplify the work.

Considering 1+2+3+4+5+6+7+ . . . +94+95+96+97+98+99+100

He noticed that the first number plus the last, 1+100, is 101.

Also, the second number plus the second-to-last number, 2+99, is 101.

AND 3+98 is 101, as is 4+97 and 5+96 and 6+95 and so on and so on.

By adding smaller numbers to bigger numbers in this way, he ended up with 50 pairs that each added to 101.  Fifty 101s is fifty times one-hundred one, which is an easy mental multiplication.  Do 5x101 and stick a zero on the end to get 5050.

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 As was Weber, Gauss was honored with a street in Gottingen being named after him, Gaussstrasse.  He has also been honored by being pictured on currency and on postage stamps.  His is buried in what was the Albani Cemetery of Gottingen; his tombstone is pictured below.

Near Gauss's grave is a pond with a fountain, which is what remains of the old moat outside the town wall.

What was the Albani Cemetery is no longer in use as a cemetery and has been re-purposed as a park for the people of Gottingen.  Given Gauss's temperament, I'm not sure how he would feel knowing people were picnicking around his tombstone.  I hadn't known this was the case until after I arrived, so it felt rather strange to me as I came reverently and on sort of a pilgrimage to pay homage to Gauss to arrive and find people sunbathing, playing Frisbee and barbecuing all around his grave, but time does move on, I suppose, and life is for the living.  Just don't be surprised if you go to see his grave as well and there's a big party going on!

David Hilbert

I had one of the best experiences of my entire sabbatical today thanks to David Hilbert and a choice he made just over a hundred years ago! Soon after the turn of the twentieth century, physicist Max Born arrived in Gottingen, and mathematicians David Hilbert and Hermann Minkowski took Born, along with their families, on a walk up to Plesse Castle for a picnic.

After having climbed there yesterday myself I have a great deal of respect for how hardy they were! It's 8 km outside of Gottingen. I took a bus the first 6 km and walked the last 2 km, and there were points where the hill was so steep that I wasn't sure if it was possible to make the walk! Once I got to the top, however, I felt it was well worth the effort. The castle itself is fascinating, and the views of the surrounding countryside and the small village below are gorgeous. The vegetation around the area is just beautiful - wooded and filled with wildflowers. The air is so fresh. It was also refreshing to visit such a place and find it more visited by locals than by tourists.

I have no idea how much or how little time Hilbert spent here over the course of his years in Gottingen, but because it was one of my favorite things of the entire sabbatical trip and because I think it gives at least some insight into his life aside from his mathematics I'm going to post many pictures of this site!

Some folks took horses up. That was smart!

Clearly many other people decided this would be a good destination - as did I - as did Hilbert, Minkowski and Born a century ago.

 I enjoyed some time to journal here.

 One can also see Gauss's tower from here - one vertex of his triangle. (See this link for details.)

The afternoon here with Hilbert and Minkowski is one that Max Born never forgot.  What he writes of it later gives some insight into Hilbert and Minkowski:

The conversation of the two friends was an intellectual fireworks display.  Full of wit and humor and still also of deep seriousness.  I myself had grown up in an atmosphere to which spirited discussion and criticism of traditional values was in no way foreign; my father's friends, most of them medical researchers like himself, loved lively free conversation; but doctors are closer to everyday life and as human beings simpler than mathematicians, whose brains work in the sphere of highest abstraction.  In any case, I still had not heard such frank, independent, free-ranging criticism of all possible proceedings of science, art, politics.

I wouldn't be surprised if Hilbert had often hiked to the castle.  He was known to enjoy walking and hiking.  He would often walk with students or colleagues in the Gottingen Forest to the east of town, pictured below.

These woods were only about one kilometer from his house at 29 Wilhelm Weber Strasse.

When Hilbert began teaching at Gottingen in 1895 all classes were held in the same building, the Auditorienhaus, but eventually a plan would be developed to create a separate mathematics institute within the university with a building of its own.

The Auditorienhaus of Gottingen University

Prior to the building of the Mathematical Institute, the third floor of the Auditorienhaus was the center of mathematical life.  Here Felix Klein, who was instrumental in bringing Hilbert to Gottingen, established a mathematical reading room called the Lesezimmer, which was entirely different from any other mathematical library in existence at the time.  Books were readily available to students - on open shelves.  Additionally Klein established a collection of mathematical models that were housed here as well.

The Mathematical Institute began with a dream of Klein's but would be brought to fruition by Richard Courant and and David Hilbert.  It was opened in 1927.  It is a T-shaped building, and on the second floor at the top of the T, in a space built specially for it, resides the collection of mathematical models and instruments.

Gottingen Mathematical Institute from the back

Gottingen Mathematical Institute from the front

Upon entering the Mathematical Institute, one sees a study area for student use - and if one looks closely, one can see that the students have a sense of humor about their august mathematical heritage.

Bust of Hilbert in the Gottingen Mathematical Institute - as decorated by students

Directly above this study area is the mathematical models collection - of which more at this link.

David Hilbert and French mathematician Henri Poincare were the greatest mathematicians of their time. When Poincare died, Hilbert stood alone as the mathematical giant of his era.  It was during the time of Klein and Hilbert that Gottingen became the most important mathematical center in the world.

David Hilbert truly was a mathematical giant.  He would dominate one branch of mathematics, creating and publishing cutting-edge work in that branch and speaking of no other area of mathematics.  But then he would suddenly enter a new area of mathematics and produce in it immediately great mature work, and before long he would move on to another entirely new area of mathematics.

On August 8, 1900, David Hilbert gave a talk at the International Congress of Mathematics in Paris, which was held at the Sorbonne.  In his talk (and the proceedings) he presented 23 problems that he felt were the most important to be solved in the twentieth century.  This single talk definintely had a powerful influence on the direction of 20th century mathematics.

The opening words of his talk are given below, along with pictures of the Sorbonne.

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be towards which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

 In January of 1930, David Hilbert reached the mandatory retirement age of 68.  He delivered a "Farewell to Teaching" speech to a lecture hall crowded with students and other professors.  He was also honored by having a street (strasse) named after him.  Upon finding out about his honor, Hilbert's wife Kathe exclaimed, "A street named after you!  Isn't that a nice idea, David?"  It is reported that David shrugged and said, "The idea, no, the execution, ah -- that is nice, Klein had to wait until he was dead to have a street named after him!"

Sadly, towards the end of Hilbert's life his country entered a very difficult time with the rise of Hitler as chancellor in 1933.  Because of Hitler's policies, many professors, especially those who were Jewish but others as well, lost their positions at universities.  Gottingen had been the mathematical center of the world, and its Mathematisches Institut had only just opened its doors in 1927, but by 1933 when Hilbert was asked how mathematics was in Gottingen now that it had been freed from its Jewish influence, Hilbert, sadly, had to reply: "Mathematics in Gottingen?  There is really none anymore."

Hilbert died in 1943, but before visiting his grave I wish to visit one more topic of importance to him, so important to him that is engraved on his tombstone.  In the mid-to-late nineteenth century there was a widely embraced reaction against the then long-standing positivism of the explanatory power of science, and that reaction was "ignoramus et ignorabimus," that is "we do not know and will not know."  It expressed a position on the limits of scientific knowledge.  Both Hilbert and Minkowski found this idea abhorrent, and in his address at the Mathematical Congress in August of 1900 at the Sorbonne in Paris he expressed that mathematical knowledge is possible with human effort.  He said:

"We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus.  For there is no ignorabimus, and in my opinion none whatever natural science.  In opposition to the foolish ignorabimus our slogan shall be Wir müssen wissen — wir werden wissen! (Our motive must be to learn -- We shall this way greatly achieve!)"

Wir müssen wissen — wir werden wissen!

We must know -- we will know!

I found the tombstone, and the engraving (though, sadly, I see that it is fading) in Gottingen's Stadtfriedhof Cemetery -

A recording of Hilbert's voice exists and can be heard on this YouTube clip. This is from a radio broadcast he gave later in life in which he reiterated his positive credo, Wir müssen wissen wir werden wissen!, which can be heard at the very end (3:56-4:01)

Wir müssen wissen wir werden wissen!

We must know; we will know!