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!

Mathematical Gottingen

Mobius strip sculpture in front of Gottingen Observatory - once home to Gauss and then Dirichlet and then Riemann

Where to begin?!

Gottingen is absolutely bursting with mathematical history.  As I mentioned in my other post on Gottingen, it included the likes of Gauss, Dirichlet, Riemann, Clebsch, Klein, Hilbert, Minkowski, Noether, and Mobius.  It was THE most important place for mathematics in the world from the time of Hilbert and Klein in the last decades of the 1800s until just before 1933 (Hitler's advent as Chancelor of Germany - at which point he got rid of the Jews and the women from Gottingen University, so there was pretty much no one left to do mathematics).

Today (Friday, May 6, 2016) I had opportunity to meet with a professor from Gottingen,  Dr. Samuel James Patterson, to whom I am profoundly grateful for his having shown me around the Mathematisches Institut and its Collection of Mathematical Models and Instruments, of which he was previously curator.  I'm also grateful for all that he shared with me in conversation regarding the personalities of the famous mathematicians of Gottingen, the history of mathematics and physics in Gottingen, the history of the university and institute there, and for directing me toward points of interest that hadn't shown up in my research before I arrived.

Dr. Samuel James Patterson of the University of Gottingen

Mathematics Institute, Gottingen - from the front - the top of the "T" (Bunsenstrasse 3-5)

Mathematics Institute, Gottingen from the back

Aside - physicists as well as mathematicians were active in this building, and it was in this area out back where, through the conversations of these physicists, that quantum mechanics was born, so I felt it quite a stunning privilege to be here for multiple reasons!

Another view from the back - bottom of the "T"

The Collection of Mathematical Models and Instruments, which is housed on the second floor of the Mathematisches Institut, displays wonderfully the history of mathematical activity in Gottingen.  It stretches back to paper models of polyhedra and other mathematical shapes hand-made by Gauss's teacher in 1780 (paper being made differently then than now!) - continues through models made by Felix Klein (1849-1925) and into the mechanical calculator era - and all the way up to the present with a fractal clock and a screen showing computer-generated models!  A small sampling is given in the pictures below - beginning with the paper models of 1780.

The famous Mobius Strip

The next two images are of models that Klein himself made in order to visualize what he was studying.  These two images are followed by a video of these types of objects as they can be studied through today's technology.

The next two images are of Klein bottles - objects with interesting topological properties.  These do not date back to Klein but were donated to the collection somewhat recently by someone who noticed they did have a Klein bottle, which seemed to them a travesty!  

And let's just continue with a few more samples that I can't resist posting:

A Riemann Surface

Calculating Devices

This piece of the collection reminds me of my office!

The collection was begun by Felix Klein prior to the construction of the Mathematical Institute Building.  At that time it could be found in a corridor on the third floor of the Auditorienhaus, the center of mathematical activity at Gottingen at the time, where the famous mathematical library, the Lesezimmer, had been founded by Klein as well.  For more information and more pictures, see my post on David Hilbert.

Gottingen University Auditorienhaus

Back to the Mathematics Institute, there are pictures of famous Gottingen mathematicians lining the halls - 

Carl Friedrich Causs

Felix Klein

Bernhard Riemann

David Hilbert

Hermann Minkowski

Emmy Noether

 - as well as interesting posters outside offices of professors.

Do you know the story behind this?  Hint: it's located above a door.

As well as the Auditorienhaus and the Mathematisches Institut, another building in Gottingen associated with mathematics and mathematicians is the observatory.  It was home (successively) to Gauss, Dirichlet and Riemann. Each would have lived in rooms on the left as you face the front (the west side).

Observatory from the back (facing south)

Inside the back courtyard - facing southeast

Wing of the observatory where the living quarters would have been (west)

There were places relating to the history of mathematics wherever I looked in Gottingen.  The two images below are of the Gänseliesel, or "Little Goose Girl" fountain, which stands in front of the Rathaus (medieval town hall). It is a tradition of Gottingen University students to kiss her upon graduating with a Ph.D.

I found in my time in Gottingen that the mathematicians and physicists of the town are both highly honored and also taken a bit for granted.  Many Gottingen streets are named after mathematicians and physicists.  Homes where famous mathematicians lived bear plaques identifying them.  Other buildings also bear plaques in honor of mathematicians.  There are also statues and busts of famous mathematicians and physicists.  (Some of the pictures following, as well as others in the post, are repeated in other posts, but I want this to be complete, so I am including them more than once.  See posts on Hilbert and Gauss for more such pictures.)

Here are some examples of these items:

Home of Hermann Minkowski - colleague and best friend of David Hilbert

On the front of the Mathematics Institute

Yet for all the honor, they are also to some degree an everyday fact of life here.  First we see the bust of David Hilbert in the Mathematics Institute, as adorned by students studying nearby.  Then we see Gauss's grave, which I approached on pilgrimage, as if it were Mecca, and then found myself having to process the fact that people were picnicking in front of it!  I wish I could remember how Dr. Patterson put it exactly when I bemoaned this state of affairs to him - something along the lines of, "We always have Gauss with us."

Dr. Patterson explained to me that where Gauss is buried, which was once the Albani Cemetery is now part of a city park, Cheltenham-Park.  It is no longer used as a cemetery, and only the most prominent stone were left.

As you can imagine the town cemeteries are filled with famous mathematicians of the past.  Alfred Clebsh is buried in a small cemetery just north of the town center, not far from the Auditorienhaus.

And then there is the Stadtfriedhof or "city cemetery" just west of the town center, which contains the graves of many mathematicians and physicists and also contains a "Nobel Circle," an area honoring all the Nobel Prize winners associated with Gottingen.  I love how so many of the stones have formulas and equations on them!

If you know of a cemetery with more famous mathematicians, physicists and Nobel Prize winners, please let me know, because I'd love to visit it.  Here is the monument to the nobelists, the "Nobel Rondell":

I spent a lot of time in cemeteries during my travels - Greyfriars Kirkyard in Scotland, Highgate in London, Pere Lachaise in Paris, and many others - all were strikingly beautiful and profoundly interesting, but I found this one to be by far the most calming and peaceful to visit.

The "Nobel Rondell" is near this pond, and many of the nobelists are buried in this area.