Carl Friedrich Gauss (1777-1855)

By Annie Pettit

EDP 180 H Fall, 2001

Introduction:

 

            Carl Friedrich Gauss is considered one of the greatest mathematicians of all time.  He is a creator in the logical-mathematical domain as he contributed many ideas to the fields of mathematics, astronomy, and physics.  Being a math education major, I have come into contact with Gauss’ work quite a few times.  He contributed greatly to the different areas of mathematics like linear algebra, calculus, and number theory.   Creativity can be seen when a person makes or discovers substantially new ideas that dramatically impact the domain in which the person is working. Gauss’ work should be considered creative because he contributed so many new theorems and ideas to mathematics, astronomy, and physics. 

Unlike some of the creators Gardner studied, Gauss seemed to be a truly decent man.  He never tried to criticize his rivals or make himself stand above the rest.  He solved problems because he loved math.  Some theorems that we credit to being solved by someone else were really discovered earlier by Gauss.  He did not publish everything because he did not have time to finish it all.  That is why I hold Gauss higher than some of the other creators we read about.  He was a decent man who worked for the love of math.  I also greatly admire his work.  Any mathematician who can prove so many different ideas in so many different areas of mathematics is truly a genius.

 

Relation to Gardner’s Triad:

 

As a child, Gauss was a prodigy.  This event happened just before Gauss turned three years old.

 

  “One Saturday Gerhard Gauss (his father) was making out the weekly payroll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention.  Coming to the end of his long computations, Gerhard was startled to hear the little boy pipe up, “Father, the reckoning is wrong, it should be….’ A check of the account showed that the figure named by Gauss was correct” (Bell 221).

 

What makes this more amazing is that nobody had taught him arithmetic.  He picked it up on his own.  Although Gauss showed great intelligence, his father refused to send him to school.  His family was very poor as his father worked as a gardener, canal tender, and bricklayer (Bell 218).  His dad wanted his son to follow in the family’s footsteps and work as a laborer.  However, his mother intervened and sent him to school when he was seven.  His teacher, Büttner, was a cold-hearted teacher who loved proving to his students how ignorant they were (Bell 221).  At the age of ten, Gauss “discovered” a formula that would change his future forever.  Büttner asked his students to add up the numbers between one and a hundred.  He figured this would keep his students busy all day.  However, Gauss noticed a pattern.  Without anyone showing him the formula [n(n+1)]/2, Gauss derived it and solved the problem quickly (Burton 510).  Büttner was so impressed by this that he bought Gauss a math book and had his assistant, Johann Bartels, work with the young boy (Bell 222).  The friendship that developed between Bartels and Gauss led Bartels to introduce Gauss to Carl Wilhelm Ferdinand, the Duke of Brunswick (Bell 224).  The Duke ended up paying for Gauss to continue his education at Caroline College, which was actually a preparatory school, and then later at the University of Göttingen.

            Like the seven creators in Gardner’s book, Gauss had similar types of relationships.  He never truly got along with his father.  His dad was a tough man who did not want his son to become educated.  Gauss was an obedient child, but he said he never really loved his father (Bell 219).  His mom, however, was a very loving person and only wanted to see him succeed.  She saw the intelligence in her son.  The person that Gauss was closest to as a child was his mom’s brother Friederich.  Friederich was an intelligent man who challenged the young Gauss (Bell 219).

            As Gauss got older, he became more of a loner.  He was more interested in his work than relationships.  His friends seemed to be people that were also interested in mathematics.  In college, he met the mathematician Wolfgang Bolyai with whom he would keep in contact with by letters until his death.  Gauss did end up marrying though.  On October 9, 1805, he married Johanne Osthof.  He wrote Bolyai telling of his happiness saying, “Life stands still before me like an eternal spring with new and brilliant colors” (Bell 243).  He had three children, but tragedy struck, and Johanne died shortly after the youngest was born on October 11, 1809.  He married again on August 4, 1810, to Minna Waldeck who happened to be a friend of his first wife.  This marriage was more for convenience as he felt his children needed a mother. 

            The period between 1806-1810 was not a good time for Gauss.  His first wife had died, and Duke Ferdinand had been killed fighting Napoleon, which meant that Gauss had to look for a real job.  Before this, the Duke had supported Gauss so Gauss was able to devote himself to research.  In 1807, he was appointed the director of the observatory at Göttingen (Bell 244).   To deal with all the tragedy that surrounded him, his work became his life. 

Gauss had published some creative ideas in the field of mathematics before his big breakthrough in 1801.  In 1796, at the age of nineteen, he discovered how to build a polygon of seventeen sides, which previously was thought to be impossible (Burton 510).  In 1801, his book Disquisitiones Arithmeticae was published.  It contained seven sections with the last one about number theory (O’Conner and Robertson).  This book caught the attention of the public, but it was too hard for anyone to really understand.  It was his work dealing with the minor planet Ceres that finally won him public honor.  His second major work was the book Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Theory of the Motion of the Heavenly Bodies Revolving round the Sun in Conic Sections) published in 1809.  We will examine these discoveries in detail a bit later.

To understand how Gauss could discover so many ideas in so many different fields, one has to look at how he worked.  He would think about mathematics and the problems he wanted to solve for days or months at a time.  While talking to people, he would often zone out because he was concentrating on a problem so hard (Bell 254).  When asked how he could accomplish so much, Gauss answered, “If others would reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries” (Bell 254).

 

Individual Level:

 

            Besides the logical-mathematical intelligence, he was strong in the visual/spatial and intrapersonal domains.  If it was not for his discovery of the seventeen-sided polygon at the age of nineteen, Gauss may have pursued a career in the study of philosophy (Bell 227).  That discovery led him into mathematics instead of philosophy although he would continue to study philosophy as a hobby throughout his lifetime.  His mathematical journal is another way Gauss’ intrapersonal intelligence comes out.  In this journal he would write out different mathematical questions he wanted to answer or different proofs that he had worked out.  Not everything in this journal was published as Gauss did not publish all his work.  This journal would serve as the proof later on that Gauss did discover many mathematical concepts that later mathematicians received credit for.  When other mathematicians would publish their work, Gauss would often tell people that he had already discovered it.  However, he never got out his journal to prove it.  He let his word be evidence enough because he had enough confidence in himself to know that he was right. Gauss also had the ability to hold an enormous amount of information in his head.  He could see things develop in his mind which would be part of the visual-spatial intelligence. 

His two weaknesses, the interpersonal and verbal-linguistic domains, were not weaknesses that hindered his creative abilities too much.  They were just not as strong as his main abilities.  Gauss was dedicated to his work so he stayed out of the mainstream public.  He had a few close friends that he kept in contact with, but he did not have a group of people around him constantly.  He also had trouble relating to some of his children (Bell 244).  That is why his interpersonal skills would not be considered strong.  As for verbal-linguistic, people had a hard time understanding his work when he published it.   He could not present his ideas very well, and a person had to be a gifted mathematician in order to understand it (Bell 230). 

            Gauss worked solely for the love and advancement of mathematics and not for the awards.  His intrinsic motivation was unbelievable.  He worked on problems that interested him and not on problems that were popular during the time.  One example is why Gauss never worked on Fermat’s Last Theorem.  In 1816, The Paris Academy offered a prize for anyone who could prove or disprove this theorem.  His friend wrote Gauss and tried to get him to work on this.  Gauss replied, “I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of” (Bell 238).  He also spent most of his time at the observatory.  Except to attend a scientific meeting in Berlin in 1836, he never slept anywhere else but under the roof of his observatory (Burton 513).

            Without the financial support of Duke Ferdinand, Gauss would not have been able to attend school.  The Duke paid his way through preparatory school and then college.  Later the Duke paid Gauss a stipend so he was able to do research.  Until the Duke’s death in 1806, Gauss did not have to worry about finding a real job.  Gauss was eternally grateful to the Duke.  He even dedicated his first book, Disquisitiones Arithmeticae, to Duke Ferdinand.  He wrote in the book, “Were it not for your unceasing benefits in support of my studies, I would not have been able to devote myself totally to my passionate love, the study of mathematics” (Burton 515).  Gauss did not have much family support.  He had a loving mother and wife, but he turned to his friends for support of his work.  He kept in touch with the mathematicians Franz Taurinus, Friedrich Bessel, and Wolfgang Bolyai using them as a sounding board for his work.

            There is evidence for the ten-year pattern that Gardner investigated.  Although there was only eight years between his locating the minor planet Ceres and his second major work Theory of the Motion of the Heavenly Bodies Revolving round the Sun in Conic Sections, his later works seem to fit the ten-year pattern.  In 1818, he started working in the field of geodesy, the study of the size and shape of the earth. He was asked to take a geodesic survey of Hanover (now a part of Germany) to connect with the Danish grid (O’Conner and Robertson).  Out of this came the invention of the heliotrope which could transmit signals by reflected light (Bell 255).  This study led Gauss to the idea of conformal mapping.  When making a map, distortions occur when transferring a surface onto a flat piece of paper.  Gauss was interested in preserving the correct angles and not distances when a map was made.  This was known as conformal mapping.  Gauss also won the Copenhagen University Prize for his Theoria attractionis which again dealt with mapping one surface onto another so “that the two are similar to their smallest parts” (O’Conner and Robertson).  In the 1830s he started work on electromagnetism.  He developed theories dealing with ideas like terrestrial magnetism and the attraction of ellipsoids (Bell 267).  In 1833, he and the physics chair at Göttingen, Wilhelm Weber, invented the first electric telegraph (Burton 514).  As a whole, his work seems to fit the pattern.  As he worked in different fields of applied mathematics, it took him almost ten years to publish a new breakthrough.

 

Domain Level:

 

         Gauss worked purely in the logical-mathematical domain.  One has to understand though that not all his contributions were to the field of pure mathematics.  Mathematics was different back then as it encompassed other fields like astronomy and physics.  Instead of having the strict divisions between disciplines as we do today, he was able to discover things in all fields with his mathematical background.  That is why his big discovery was actually in astronomy although it was mathematically based.

In 1801, an Italian astronomer Piazzi had discovered a minor planet named Ceres.  Ceres disappeared behind the sun and nobody could locate it again.  Piazzi did not have enough time to collect a lot of data on Ceres before it disappeared so calculating its orbit proved to be a big challenge to the scientific community.  Newton had even said that these problems were among the toughest to solve in mathematical astronomy (Bell 241).  Even with the meager data collected, Gauss was able to calculate the orbit of Ceres, and when the planet reappeared, it was in the exact spot Gauss had predicted.  This won him fame in the eye of the public, but this discovery was not without criticism.  Prominent men criticized him for wasting his time calculating the orbit of a minor planet.  After all, that would not help the advancement of their city.  The calculations he used later became known as the Gauss-Jordan elimination and the method of least squares.  Gauss-Jordan elimination is an algorithm for solving systems of linear equations.  An example of a system of five linear equations would be the following:

 

5a+2b+5c+7d+3e=1

2a+4c+3d=5

4b+9c+6d+e=2

6b+2e=3

9a+7b+5c+6d+8e=1

 

In solving Ceres’ orbit, Gauss had a system of seventeen linear equations that he had to calculate (Bretscher 23).  The method of least squares is “minimizing the sum of the squares of the components” or finding the smallest distance between two vectors (Bretscher 222). 

Theory of the Motion of the Heavenly Bodies Revolving round the Sun in Conic Sections was the second book written by Gauss. The first volume deals with differential equations, conic sections and elliptic orbits while the second examines in more detail the estimation of a planet’s orbit (O’Conner and Robertson).  Differential equations deals with systems of equations, but instead of having just variables, derivatives of variables are involved. 

Obviously, Gauss’ work dealt almost purely with numbers.  This was good as Gauss had a hard time communicating his ideas to the general public.  Although he was well read, his verbal-linguistic intelligence was not one of his strongest.

The fields of mathematics and astronomy were pretty established when Gauss calculated the orbit of Ceres.  This calculation did not dramatically change the fields or cause other mathematicians to change their areas of study to astronomy.  What he did was solve a problem that had perplexed scientists for years.  The fact that he did it with so little data was amazing.

Field Level:

            Besides his uncle Friederich and Johann Bartels helping him as a young child, Gauss never really had any mentors.  He was always ahead of his teachers in the field of mathematics and preferred to work out things on his own.  He had friends who he discussed math with like the professor Johann Friedrich Pfaff, but he never had anyone guide him in the field of mathematics. 

Gauss disliked criticism or controversy.  Thus he tried to steer clear of any confrontation with his rivals.  When Gauss said he had discovered something first, he never made it a big deal.  One example of this would be with the mathematician Legendre who claimed that Gauss stole the method of least squares from him since Legendre had published it in 1806.  Although Gauss used this method in 1801 to calculate Ceres’ orbit, he did not publish it until 1809.  Gauss refused to get into an argument with Legendre.  Gauss had written Olbers in 1802 with the idea of least squares, and he had the evidence to show Legendre.  Gauss, however, did not want to draw the argument out.  He let his word stand on its own accord (Bell 259).    

One criticism Gauss’ peers had was how he only published works that had been perfected.  Other mathematicians asked him again and again to relax a bit and publish works even if they were not completely finished as this would help advance the mathematical field more quickly.  Gauss refused to do this (Bell 230).

Another criticism of Gauss is that he did not support the younger mathematicians who followed him.  When they published something, he never showed his praise.  For example, when Cauchy published his theory of functions of a complex variable, Gauss did not say one thing about the work (Bell 260).  What people did not realize was that Gauss had already discovered the complex variables.  He just never got around to publishing it.

            Gauss’ work never drew much political controversy.  To understand why, one has to look at what Gauss published and his personality.  Gauss never published any work that was not completely perfect.  He did not want anyone adding or taking away from his work (Bell 230).  Thus, when he published something, it was so complete that nobody could find any holes in it.  However, that prevented him from printing some mathematical ideas that he discovered.  Many of the ideas that were discovered after him were later found to have been discovered first by Gauss.  He just did not get around to publishing the discoveries because he did not have time to finish them or for fear of criticism.  This is where his personality came in.  Gauss disliked controversy.  Therefore, when he discovered a non-Euclidean geometry, he did not publish it for fear of criticism and/or controversy.  Everything that was understood in the world at that time was explained by Euclidean geometry.  For someone to imagine a geometry out there based on something else would have shocked the world.  That is why Gauss never published it.  He wrote to his friend Franz Taurinus in 1824 that he felt that telling the public about this new discovery would subject him to ridicule (Burton 550).  The credit for non-Euclidean geometry goes to Lobachevsky and John Bolyai, the son of Gauss’ longtime friend. Gauss receives some recognition for laying the foundation of non-Euclidean geometry as his journal showed evidence of Gauss’ research.

            To understand how important Gauss was to the field of mathematics, one has to only look to Laplace’s comment on Gauss.  Alexander von Humboldt asked Laplace, a great mathematician in his own right, who the greatest mathematician in Germany was.  Laplace said it was Pfaff.  When Humboldt asked why he did not answer Gauss, Laplace replied, “Gauss is the greatest mathematician in the world” (Bell 242).  Gauss spent most of his adult life at University of Göttingen.  Although he would have been a good candidate for a political office as he followed politics pretty closely, he was content to stay at the university and do research.  When he died in 1855, he was in an elite class with Archimedes and Newton as the greatest mathematicians to have ever lived (Bell 218).

 

Conclusion:

 

         Gauss fit Gardner’s model pretty well.  If Gardner had focused on individuals that made their creative breakthroughs in the early 1800s, he would probably have included Gauss.  Gauss fit the childhood profile as the other creators in the fact that he was closer to someone other than his parents as he related best to his uncle Friederich.  Gauss was also a loner.  He did not have many close friends.  He preferred to immerse himself in his work than to form relationships.  Again, most of the seven creators were considered loners.  Although Gauss’ second work did not occur exactly ten years later, it was pretty close.  He also continued to produce works and even had a breakthrough thirty years after the original with his work of electromagnetism and the electric telegraph.

            What did not fit Gardner’s model was how Gauss’ big breakthrough was not an earth-shattering discovery.  Although the concept of calculating a planet’s orbit was puzzling to the scientific community, they knew it could be done.  They just did not know how to go about it.  Therefore, it was not a huge shock to mathematicians and scientists when Gauss figured it out.  If Gauss had been braver and published his ideas on a non-Euclidean geometry, then he would have fit Gardner’s model almost perfectly.  Instead he chose to publish works that would not raise a lot of political controversy.  Although Gauss is considered one of the greatest mathematicians of all time, he would have been in a class by himself if he would have published everything he had discovered.

Works Cited

Bell, E.T. Men of Mathematics. New York:  Simon and Schuster, 1986.

Bretscher, Otto. Linear Algebra with Applications. Upper Saddle River, New Jersey: Prentice-Hall, Inc., 1997.

Burton, David M. The History of Mathematics, an Introduction. Newton, Massachusetts: Allyn and Bacon, Inc., 1985.

O’Conner, J.J. and E.F. Robertson. “Johann Carl Friedrich Gauss.” (Dec. 1996). 26 November, 2001

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html