Each university and each department develops a peculiar kind of folklore -- anecdotes about those of its graduates (or dropouts) that somehow managed to become famous (or notorious). Very often, there is an element of glee to these stories: "Well, he may now be a government minister, but I flunked him in calculus!" And also very often, it is impossible to tell the truth from the legend.
 

When I was a math student at Münster, Germany in the 1980s, such anecdotes centered mainly around two people: Gerd Faltings, the first and only German to win the Fields Medal, mathematics' equivalent of the Nobel prize; and Karl Weierstraß, the man who (besides many other mathematical accomplishments) introduced   and    into calculus.

Weierstraß had been a student at Münster in the 1830s and 1840s. There was no one around anymore who knew anybody who had known anybody who had known anybody who had known Weierstraß, but this didn't prevent the folklore from blooming. According to his legend, Weierstraß flunked out of law school because he spent most of his time there drinking beer and doing mathematics).

Then he worked for more than ten years as a school teacher in remote parts of Prussia, teaching not only mathematics, but also subjects like botany, calligraphy, and physical education. Finally, when almost 40 years old, he became a famous mathematician, and was eventually appointed a professor at Berlin --- without ever having received a PhD. This story may sound wild, and in some ways it simplifies the facts, but it is not far from the truth.
 

Karl Theodor Wilhelm Weierstraß was born on October 31, 1815, in the village of Ostenfelde, which was located in what was then the Prussian province of Westphalia. A street and an elementary school in Ostenfelde are named after him, and his birth house --- still occupied today --- is listed in a local tourist guide. His father, who worked for Prussia's customs and taxation authorities, was sent from one post to the next within short periods of time.

For the first 14 years of Karl Weierstraß' life, his family was more or less constantly on the move. In 1829, Karl's father obtained an assistant's position at the tax office in the city of Paderborn (also in Westphalia), and the family could finally settle down.

Young Karl enrolled at the local Catholic Gymnasium in Paderborn, where he excelled not only in mathematics, but also in German, Latin, and Greek. Not only was he a strong student, he was also quite capable of putting his brains to work on much more practical matters. At age 15, Karl contributed to his family's income by doing the bookkeeping for a wealthy merchant's widow.

Throughout his life, Weierstraß Senior suffered from the knowledge that he did not have the right education to rise to a rank in the Prussian civil service that would have better suited his abilities. Instead he had to content himself with relatively low-level positions, not very challenging and not very well paid. Like many a father in this situation, he was determined to prevent such a fate befalling his bright eldest son. When Karl graduated in 1834, his father decided to send him to Bonn to study  Kameralistik  (a combination of law, finance, and administration).

Being a dutiful son, Karl went  ...   and did all he could to sabotage the life his father had planned for him. He joined a  schlagende Verbindung, a kind of student fraternity typical of German universities in the 19th century.

Besides keeping the brewing industry busy, fraternity members engaged in a peculiar ritual: the  Mensur, a swordfight with a peculiar twist. Unlike in today's athletic competitions, the students fought with sharp sabers. They wore protective gear which covered most of their bodies --- except the cheeks. During a Mensur, the opponents tried to inflict gashes on one another's cheeks. The scars were borne with pride as signs of honor and manhood. Almost two metres tall, quick on his feet, and with strong arms, Karl Weierstraß was a fearsome swordsman. His face remained unscarred, and after a while nobody was keen on challenging him anymore.

Having escaped from under his father's tutelage, he spent his years at Bonn drinking beer and wielding the saber - and seriously studying mathematics. Although he was not enrolled in mathematics, he read some of the most advanced math books of his time. In 1838, when it was time for him to take his exams, he simply dropped out.

His family was desperate. They had made considerable financial sacrifices to secure a better future for Karl, who had let them down. Having wasted four years of his life, he needed a breadwinning degree and fast. So, in 1839, he enrolled at the Akademie in Münster, the forerunner to today's university, to become a secondary school teacher. Although this was not really a university, but rather a teacher training college, they had one good mathematician teaching there - Christoph Gudermann. He is said to have been an abysmal teacher: very often, he had just one student sitting in his class - Karl Weierstraß.

In 1840, Weierstraß graduated. His thesis was so good that Gudermann believed it to be strong enough for a doctoral degree. However, the Akademie was not really a university; it did not have the right to grant doctorates. So, instead of receiving a doctorate and starting an academic career, Weierstraß left the Akademie as a mere school teacher.

His first job (probationary) was in Münster. One year later, he was sent to Deutsch Krona, in the province of West Prussia as an auxiliary teacher then in 1848, to Braunsberg in East Prussia. Of course he taught mathematics, but also physics, geography, history, German, and --- believe it or not --- calligraphy and physical education. Besides the demands of working full time as a teacher and having a social life, he found time to do research in mathematics.

During his time in Braunsberg, he published a few papers in his school's yearbook. High school year books are not exactly where people look for cutting edge research in mathematics, and so nobody noticed them. Then, in 1854, he published a paper entitled ``Zur Theorie der Abelschen Functionen'' in a widely respected journal. I won't even make an attempt to explain what it was about. But unlike his previous work, this one was noticed.

It dawned to mathematicians all over Europe that the man who was probably the leading analyst of his day was rotting in a small East Prussian town, spending most of his time teaching youngsters calligraphy and physical education. On March 31, 1854, Weierstraß finally received a doctorate, an honorary one from the university of Königsberg.

In 1856, he accepted a position at the Gewerbeinstitut in Berlin, an engineering school, and a year later he joined the faculty of the University of Berlin as an adjunct professor. As a teacher, he attracted large audiences. Often, he taught in front of more than 200 students. In 1869, when he was almost 50 years old, Weierstraß was appointed full professor at the university of Berlin. In 1873 and 1874, he was Rektor magnificus of the university; in 1875, he became a knight of the order "Pour le Mérite" in the category of Arts and Sciences, the highest honor newly unified Germany could bestow upon one of its citizens; and, in 1885, on the occasion of his 70th birthday, a commemorative coin was issued in his honour.

The years of leading a double life as a secondary school teacher and a mathematical researcher took their toll on Weierstraß' health. A less vigorous man would probably have collapsed under the double burden much earlier. In 1850, Weierstraß began to suffer from attacks of dizziness, which culminated in a collapse in 1861. He had to pause for a year before he could teach again, and he never recovered fully. In 1890, at age 75, Weierstraß retired from teaching because of his failing health. The last years of his life were spent in a wheelchair. In 1897, he died.

Weierstraß published few papers - he was very critical toward his own work. But although he was a brilliant researcher, the greatest impact he had on mathematics was as a teacher. At Berlin he repeatedly taught a two-year course on analysis, the predecessor of all modern introductions to calculus and analysis. Although he never wrote a textbook, notes taken in class by his students have survived and convey an impression of his lectures.

Perhaps the longest lasting legacy of those lectures is their emphasis on rigor. When calculus was created in the 17th century, mathematicians did not worry about rigorously proving their results. For example, the first derivative  dy/dx  of a function  ()  was thought of as a quotient of two "infinitesimals", (i.e. infinitely small quantities    and  ). Nobody could really tell what infinitely small quantities were supposed to be, but mathematicians then didn't really care. The new mathematics enabled them to solve problems in physics and engineering which had been beyond the reach of human mind before. So why bother with rigor?

In the 18th century, mathematicians went so far as to proclaim that rigor was for philosophers and theologians, not for mathematicians. But with the lack of rigor, contradictory results cropped up with disturbing frequency --- people arrived at formulae that were obviously wrong. And, if a particular formula determines whether or not a bridge collapses, you don't want it to be wrong.

Weierstraß realized that if calculus was to rest on solid foundations, its central notion, that of limit, had to be made rigorous. He introduced the definition which (essentially) is still used today in classrooms:

A number    is the limit of a function  ()  as    tends to    if, for each  > , there is  >   such that  |() |   for each    with  | | .

Students may curse it, but it will not go away.

Weierstraß was not only an influential lecturer, but also one of the most prolific advisors of PhD theses of all time. There is a database on the Internet, The Mathematics Genealogy Project, which lists 31 PhD students of Weierstraß and 1,346 descendants (i.e.  PhDs of PhDs of PhDs etc.) of Weierstraß. Interestingly, the two former students who generated the most folklore weren't his students in a technical sense.

Sofya Kovalevskaya was a young Russian noble woman who had come to Germany to study mathematics. This alone was not a small feat at a time when the very idea of a woman receiving a university education was revolutionary. For two years, she studied at Heidelberg, where authorities would not let her enroll officially, but eventually allowed her to attend lectures unofficially (provided the instructor did not object). Then she moved to Berlin to work with Weierstraß only to find that she was not even allowed to sit in lectures.
 

This prompted Weierstraß, by all we know a politically conservative man, to tutor her privately. Since Kovalevskaya could not receive a doctorate from Berlin, Weierstraß used his influence to persuade the University of Göttingen to award her the degree in 1874. She spent the following nine years jobhunting. Being a woman didn't help. The best job she could find was teaching arithmetic at an elementary school. Finally, in 1883, she was offered a professorship at Stockholm, where she worked until her death in 1891 at the age of 41.

 
Weierstraß and Kovalevskaya stayed in touch throughout her mathematical career. After her death, Weierstraß destroyed their correspondence. This fact, along with Kovalevskaya's striking beauty, gave rise to innuendo that she may have been more to Weierstraß (who never married) than just a student. May have been - but we don't know.
 

Gösta Mittag-Leffler, another of the great mathematicians protegées, was also not Weierstraß' student strictly speaking . Already enrolled at the University of Uppsala, Sweden, he came to Berlin in 1875 to attend Weierstraß's lectures, which turned out to have an enormous impact on his mathematical development. He then returned to his native Sweden, where he received his doctorate.

Over the years, Mittag-Leffler became indisputably the most influential mathematician of his time in Sweden.

 
He made use of his clout to overcome the obstacles faced by Sofya Kovalevskaya regarding her appointment at Stockholm. What Mittag-Leffler is most famous for, however, is not a mathematical accomplishment, but a piece of mathematical folklore. To this day, mathematicians suffer quietly from the lack of a Nobel prize, and, some say, Mittag-Leffler is to blame - according to legend, the first version of Nobel's will mentioned a prize in mathematics. Then, Nobel found out that his wife had had an affair with Mittag-Leffler. Infuriated that his wife's lover could well be the first prize winner, Nobel changed his will and removed the math prize. That's a fine piece of juicy folklore, but nothing more - like Weierstraß, Nobel was a lifelong bachelor.