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James Stirling's Methodus Differentialis: An Annotated Translation of Stirling's Text

Ian Tweddle
Springer Verlag
Publication Date: 
Number of Pages: 
Sources and Studies in the History of Mathematics and Physical Sciences
[Reviewed by
Fernando Q. Gouvêa
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Ian Tweddle's new edition of James Stirling's Methodus Differentialis is a useful and important book. This is an annotated translation of the original text, and hence perhaps a little more friendly to non-historians than the Beeley-Scriba edition of Wallis' letters. Tweddle has translated the Latin and added many notes. Much of Stirling's book deals with numerical methods for finding the sums of series and other function values; Stirling gives no error estimates for his methods, so Tweddle has given these in his notes.

The original title of Stirling's book (originally published in 1730) can be translated as "The Method of Differentials, or, a Treatise on Summation and Interpolation of Infinite Series." That is a good description of what is in the book. Stirling's Formula is here, as are the original appearances of the Stirling numbers and of Stirling's interpolation formula. There is also lots of stuff on transformations of series (for example, to accelerate their convergence). The blurb on the back cover describes the book as a classic of numerical analysis, and that seems exactly right.

This is another expensive book, a bit too expensive in fact, which is a pity. At a lower price, it might have reached a bigger public. (Working through Stirling might make a very interesting — and very challenging — undergraduate project, one that would make sense to students interested in "getting a number out" of all the mathematical theory.) As it is, I hope that all serious libraries will consider buying a copy.

 Fernando Q. Gouvêa is a wannabe historian.

Introduction.- Background.- Some Mathematical Points.- Summary of the Contents of Stirling's Text.- Stirling's Principal Calculations.- Stirling's Text in Translation: Preface; Introduction; Part I: On the Summation of Series; Part II: On the Interpolation of Series.- Notes.- Appendix: Stirling's Letter to De Moivre Dated 17th June 1729.- References.- Index.