**Christoff Rudolff** was a German mathematician whose book *Coss* is the first German algebra book.

- It is possibly that this is a relative of Rudolff's or even Rudolff himself.
- Schreyber taught at the University of Vienna between 1517 and 1521, so we know that Rudolff must have studied there for some period before 1521.
- Rudolff's book Coss, written in 1525, is the first German algebra book.
- At this time Brixen was controlled by the Habsburgs and this dedication makes one wonder whether Rudolff had spent some time there.
- Although this is the first German algebra book, containing some important innovations, there were already a number of algebra books (some only existing in manuscript) which Rudolff had studied before writing his text.
- We will return to look at the contents of Coss in a moment, but first let us look at Rudolff's other works.
- In 1526, the year after Coss appeared, Rudolff produced Künstliche Rechnung mit der Ziffer und mit den Zahlpfennigen Ⓣ(Calculating sums with the number and the number of pennies).
- In 1530 Rudolff produced the book Exempelbüchlin Ⓣ(Example booklet) which has much in common with the third part of his earlier work and contains 293 problems.
- His notation for decimals is not very different from that used today; instead of the modern "decimal point", Rudolff uses a bar to separate the integral and fractional parts.
- If any particular individual were to be named as having the best reason to be called the inventor of decimal fractions, Rudolff would seem to be the man, because he apparently knew how to operate with these forms as well as merely to write them, as various predecessors had done.
- Let us return to Rudolff's Coss and, first, give its full title: Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre so gemeinlicklich die Coss genent werden Ⓣ(A clever way of calculating sums by the artful rules of algebra commonly called 'the thing').
- We noted above some of the manuscripts on algebra which Rudolff had read before producing his book and in fact many of his fellow mathematicians criticised him for 'stealing' his examples from earlier texts.
- It was standard practice at this time, and for many years after this, to use examples from earlier works and Michael Stifel strongly defends Rudolff from these attacks.
- It is reasonable to believe that Rudolff's fellow mathematicians were upset at this book because it was written in German rather than Latin.
- There are two parts to the book: the first part contains twelve chapters and this material, writes Rudolff, must be mastered by the reader before he progresses to a study of algebra which comprises the second part of the book.
- Returning to the first edition, Rudolff computes with expressions involving square roots.
- Rudolff goes on to look at adding, subtracting, multiplying and dividing polynomial expressions.
- Earlier works on solving equations had presented the reader with 24 different cases but Rudolff reduces this to 8 cases.
- In some of the problems Rudolff introduces a second unknown, q (for 'quantitas').
- For several such problems concerned with "splitting the bill" (Zechenaufgaben) Rudolff supplied all the possible solutions.
- Rudolff does not work out their solutions because, as he stated, he wanted to stimulate further algebraic research.
- Another important feature of Coss is that Rudolff uses letters to represent numbers.
- Adding a diagonal stroke to the points used by earlier cossists, Rudolff introduced ...
- In brief, Rudolff's role in the development of mathematical studies in Germany was analogous to that of Fibonacci in Italy.
- As to his influence on later mathematicians, we note that Michael Stifel brought out a new addition to Rudolff's Coss in 1553, adding some important material of his own.
- When Euler was learning mathematics he made a careful study of Rudolff's work.

Born 1499, Jauer, Silesia (now Jawor, Poland). Died 1543, Vienna, Austria.

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Origin Poland

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive