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BOOK REVIEWS BEFORE CALCULUS Janet Beery And

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BOOK REVIEWS BEFORE CALCULUS Janet Beery And BOOK REVIEWS BEFORE CALCULUS Downloaded from rsnr.royalsocietypublishing.org on January 2, 2011 304 Book Reviews 10, .... These are the triangular numbers in The edition offers a facsimile reproduc- a narrower sense, known since antiquity: tion of 40 pages of the manuscript source, each entry represents the number of dots in with an explanatory commentary on facing an equilateral triangle evenly filled with pages. No transcription is needed, because dots. And Harriot continues to produce Harriot’s writing is clearly readable in this rows of sums of the entries of the previous treatise, and his algebraic notation is row, the next row being 1, 4, 10, 20, .... nearly modern. Nevertheless the editors The numbers in such a table may be referred transcribe selected formulae into moder- to as triangular in a broader sense, and it nized notation to facilitate reading. Latin is in reference to this that Harriot’s prose, which is very rare, is transcribed treatise is headed a ‘doctrine of triangular and translated into English. numbers’. The edition also offers a 53-page essay Harriot then passes on to tables with that introduces the topic of triangular num- sequences whose first element may be any bers, difference tables and interpolation natural number. On reading these tables and thus provides a helpful preparation for backwards, that is, from a sequence of reading the manuscript itself. The essay sums to the sequence of its summands, the further discusses in considerable detail the sum table becomes a difference table. history of the reception and influence of Such difference tables can be produced Harriot’s manuscript treatise. The authors starting from any table of functional analyse the work on difference tables and values, and in particular also values of trans- interpolation by Nathaniel Torporley, Henry cendental functions such as the sine, tangent Briggs, Walter Warner, Charles Cavendish, and logarithmic functions. Although in gen- John Pell, John Collins, Nicolaus Mercator, eral the continuous taking of differences of Isaac Newton and James Gregory, differences will not yield a sequence of thereby demonstrating that difference tables constant differences, one can use such were a much-discussed theme of English tables for approximate interpolation by sti- seventeenth-century mathematics and pulating a given sequence of differences to uncovering a network of informal be constant. communication between mathematicians. In Harriot then describes the entries of particular, the authors reveal a hitherto difference tables in terms of the first unrecognized relation between Mercator elements of each sequence, using his con- and Warner. venient algebraic notation. By considering Historians of mathematics will welcome partial tables whose starting sequence con- this book as providing new insights into sists of only every nth element of the start- the development of mathematics before the ing sequence of a given difference table, invention of the calculus and as a model he is able to derive general formulae of for the publication of historical sources on how the first elements of the original mathematics. Dealing with a neat, easily table depend on those of the partial one. comprehensible aspect of mathematics in a By constructing a difference table using historically exciting period, the book has these formulae he is able to infer interp- the potential to raise the interest of mathe- olation formulae expressing new functional maticians in history and of historians in values in terms of given ones and their mathematics. differences, differences of differences, and so on. These interpolation formulae he doi:10.1098/rsnr.2010.0016 calls ‘Magisteria’. Published online 21 April 2010.
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