You are here

Mathematical Treasures - Widman's Commercial Arithmetic

Author(s): 
Frank J. Swetz (The Pennsylvania State University)

Published in Leipzig in 1489, Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften (Nimble and Neat Calculation in all Trades) was the first printed commercial arithmetic book to appear in Germany. The author was Johannes Widman (ca.1460–1498), a graduate of the University of Leipzig who returned to the university to teach mathematics. Widman introduced the plus sign “+” and the minus sign “” into the context of European mathematics. The title page for the 1508 edition is shown below, depicting merchants at a counting table. It is difficult to discern if the man at lower right is working on an abacus or writing on a slate board.

Title page for 1508 edition of Widman's Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften.

On the pages below, Widman provided three examples of galley division:

Left: 9803617524 ÷ 45678 = 214624 + 22452/45678

Right: 161143442 ÷ 2159 = 74638; 3900994564 ÷ 3986 = 978674.

Folio 18 from 1508 edition of Widman's Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften.

The six pages below provide some situational word problems, including a problem involving a courier and another involving the will of a dying man, a will that requires a mathematical interpretation.

Folio 96 from 1508 edition of Widman's Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften.

Folio 97 from 1508 edition of Widman's Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften.


Folio 98 from 1508 edition of Widman's Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften.

These images from its George Arthur Plimpton Collection are presented through the courtesy of the Columbia University Libraries.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasures - Widman's Commercial Arithmetic," Convergence (January 2021)

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University