**Johannes Widman** (whose name is also written as **Weideman** or **Wideman)** attended the University of Leipzig and his name appears on the list of those registered for the winter semester of 1480 as 'Iohannes Weideman de Egra'. He graduated with a first degree in 1482 and he continued to study for a Master's degree being allowed to live outside the dormitory. His Master's Degree was awarded in 1485 and he then taught at the University of Leipzig on the fundamentals of arithmetic, computation on lines, and algebra. His lectures were advertised, and students were invited to attend.

Widman's 1486 algebra lecture is the first to be given in Germany on that topic, and amazingly still survives in a notebook of a student who attended. Widman used Cossist notation, as was usual at that time, discussing 24 different types of equations. The approach is unusual in that it contained symbols for addition, subtraction, and square roots. He considered computation with irrational numbers and polynomials to be part of algebra, preparing his students for this study by first introducing them to fractions and proportion.

It is, however, for an early arithmetic book *Behende und hupsche Rechnung auf allen kauffmanschafft,* published in German in 1489, that Widman is best remembered. This book has become famous since it contains the first appearance of + and - signs for addition and subtraction. It is an early example of a printed arithmetic book and it is better than those before it in having more examples and also a wider range of useful examples. The book consists of three parts: the first section is on counting with whole numbers, the second is on proportion, while the third section is on geometry. A I Volodarskii, reviewing [4], writes:-

Around the end of the Medieval period, changes and modifications took place in the development of mining and metallurgy which had a great influence on economic life. All these led to an increase of monetary circulation and to changes in the methods of practical arithmetic. At the first stage, trade was reduced to barter. The mathematical rules and practical problems concerning barter were given ... in the treatises of Widman, Pellos, Pacioli, De la Roche, Ries, Rudolff, Tartaglia, and Cardan. Gradually barter was ousted from everyday practice and the corresponding mathematical problems also vanished from the training aids. ... another merchant method for practical calculation, the so-called table(or tabular)method ... was also described by Widman. ... The chain rule was in use longer than other merchant methods of calculation. It can be found in[Widman's book].

The popularity of Widman's book can be seen from the fact that it was reprinted in 1508 in Pforzheim, in 1519 in Hagenau and in 1526 in Augsburg. Notice also that being reprinted in several different places indicates that it was popular over a large part of the German speaking world. After 1526 the work was superseded by an arithmetic book by Adam Ries and then by further works from other authors.

There are no direct references to Widman after 1489 but it is believed that he was still working on mathematical topics as late as 1498. In fact in [2] there is a discussion whether Widman stayed in Annaberg around 1500 which would be relevant since Adam Ries lived there around that time. Widman was probably the author of the book *Algorithmus Linealis* published in Leipzig in 1489. This book is the oldest printed work containing instructions for counting with the help of an abacus.

Widman's importance is that he provided one of the first practical applications of mathematical knowledge which, because of the recent invention of printing, was made accessible to a very much wider circle of readers than had ever been achieved before.

**Article by:** *J J O'Connor* and *E F Robertson*

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