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Jost Bürgi's Aritmetische und Geometrische Progreß Tabulen (1620)

Kathleen Clark
Publication Date: 
Number of Pages: 
Science Networks Historical Studies 53
[Reviewed by
P. N. Ruane
, on

The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, foreshadowed it or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought’

                        Lord Moulton, Napier Tercentenary Memorial Volume, London 1915


Logarithms were devised by the independent efforts of two men: John Napier (1550–1617) and Jost Burgi (1552–1632); they completed their individual versions of logarithms before index notation came into common mathematical use. Consequently, neither of them made overt use of the relationship \(y=10^x \Leftrightarrow \log_{10}y=x\) (although Bürgi’s method is almost amenable to it).

Napier’s approach, described in an appendix in this book, was geometric; his tables were published in 1614. The key to Burgi’s tables was his use of arithmetic and geometric series (Aritmetische und Geometrische Progress) which is referred to in the title of his manuscript. And although his tables were printed six years later than those of Napier, he may have developed his ideas some time before Napier did. But it is the close examination of Bürgi’s tables (which were actually antilogarithmic) that occupy the bulk of this book.

Kathleen Clark says that she isn’t a traditionally trained historian, and her interest in this subject arose from her work with high school teachers on ways to teach logarithms and logarithmic functions using an historical perspective. That’s one excellent reason for the production of this book, and the fact of its originality is another.

The narrative centres upon a facsimile copy of Jost Bürgi’s rarely available manuscript, which consists of 48 pages of trigonometric and logarithmic tables, preceded by 23 pages of Bürgi’s handwritten set of instructions for their use. These are transcribed into more readable German, and then, with no seeming misrepresentation of Bürgi’s tone and style, they are translated into English (by the Kathleen Clark, I presume).

The author’s most notable mathematical input is the commentary that she has inserted into the translated version of Bürgi’s manuscript. In the process, she explains his mathematical ideas in modern notation and clarifies the means by which the tables can be used. Her historical analysis partly concerns the relationship between two known manuscripts of Bürgi’s tables — one of which is housed in the University of Graz (Austria); the other resides in Gdansk (Poland).

To conclude: if Kathleen Clark considers herself untrained in the historical arts, she should make no apologies. Her book forms a unique contribution to the history of mathematics, and I don’t see how she could have improved upon the standard of scholarship that is so evident within its pages.

Peter Ruane only ever taught Napierian logarithms; but he also used Napier’s rod in the teaching of multiplication in elementary school. During school and university teaching, the mathematics ran from elementary arithmetic to transfinite arithmetic. 

See the table of contents in the publisher's webpage.