You are here

Mathematical Treasure: Benjamin Bramer's Applied Geometry

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

Benjamin Bramer (1588-1652) was a German mathematician, architect and instrument maker. At an early age, Bramer was adopted by the Swiss mathematician and astronomer Jost Bürgi (1552-1632). Bürgi mentored his young charge in mathematics and scientific skills. In 1646-1647, Bramer published Apollonius Cattus, oder geometrischer Wegweiser. The last two words of the title mean "geometric signpost" and this earlier work can be viewed in its entirety on the e-rara website. This book was republished in 1684 as Apollonius Cattus oder … Geometriae, and pages from this latter book are featured here. In these books, Bramer stressed the applications of geometry and the use of measuring instruments.

Title page of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Bramer used the frontispiece as a tribute to Bürgi, who had become his brother-in-law as well as his mentor. An excerpt of Bürgi’s work is also included as a section in the text.

Frontispiece from Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Bramer’s experience as an architect and builder of fortifications influenced his dynamic view of geometric objects. He often projected cross-sections to understand shapes.

Page 1 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Page 2 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Page 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Page 8 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

The opening page of Bürgi’s contribution:

First page of the third part of of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

The following five images provide a sampling of the surveying and measuring situations considered in Bürgi’s presentation. Figure I illustrates the measurement of inaccessible heights.

Figure 1 from Part 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Plate 8 demonstrates the triangulation of an open field.

Plate 8 from Part 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Plate 15 shows how to utilize available heights to obtain distance measures to remote structures.

Plate 15 from Part 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Plate 19 shows the surveyor how to measure (or estimate) the height of a cloud. The line that begins "Die höhe …" on page 20 (shown below) translates literally as "The height of a Cloud from the Earth to measure."

Page 20 from Part 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

Plate 19 from Part 3 of Apollonius Cattus oder ... Geometriae by Benjamin Bramer and Jost Burgi, 1684

The images above are presented courtesy of ETH-Bibliothek Zürich and the work may be viewed in its entirety via e-rara.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Benjamin Bramer's Applied Geometry," Convergence (August 2018)

Dummy View - NOT TO BE DELETED