You are here

Mathematical Treasures - Simon Jacob's Rechenbuch

Frank J. Swetz and Victor J. Katz


Tjitle page

This is the title page of the "New and Fully Revised" Rechenbuch of Simon Jacob (d. 1564), one of the best-known Rechenmeisters of the sixteenth century. The book was first published in 1560, but this illustration is from the 1565 edition. Other editions were printed up to 1600. The book was a commercial arithmetic text, in which the basic laws of arithmetic were applied to many practical problems. A wide variety of mathematical instruments are illustrated on this page.


This is the title page of the 1599 edition of Jacob's first book, the Rechenbuchlein, originally published in 1557. It also went through several editions.

Cube root extraction

This illustration and those that follow are all from the 1565 book. On this page, Jacob has sketched the decomposition of a solid as an illustration of his method of calculating cube roots.

Description of cube roots

On the left-hand page there is part of the computation of the fourth root of 76656. Note that the answer is not a whole number. On the right-hand page, Jacob begins his description of the general procedure for calculating higher roots.

Pascal triangle

Since the general method of calculating roots involves what we call the Pascal triangle, Jacob gives this triangle through row 12 on page 104.

More on roots

Jacob continues his description of the method of finding roots. On the right-hand page, he explicitly states the binomial theorem in several cases.


Calculation of a fourth root

On the right-hand page, Jacob illustrates the calculation of the fifth root of 119643398733624. The answer is 654.

Index to Mathematical Treasures

Frank J. Swetz and Victor J. Katz, "Mathematical Treasures - Simon Jacob's Rechenbuch," Convergence (January 2011)

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University