Downloaded from rsnr.royalsocietypublishing.org on January 2, 2011
Notes Rec. R. Soc. (2010) 64, 303–311
BOOK REVIEWS BEFORE CALCULUS
Janet Beery and Jacqueline Stedall (eds), Thomas Harriot’s doctrine of triangular numbers: the ‘Magisteria Magna’. European Mathematical Society, Zurich, 2009. Pp. viii þ 137, £55.00 (hardback). ISBN 978-3-03719-059-3.
reviewed by Matthias Schemmel*
Max Planck Institute for the History of Science, Boltzmannstrasse 22, 14195 Berlin, Germany
In the history of the mathematical sciences difference tables to interpolate between it is fascinating to see how new challenges given values of a function. The book thus bring about new solutions. In early modern shows what was possible in dealing with times, mathematicians were increasingly functional relations before the advent of confronted with functional relations the calculus. between quantities. These relations, in Harriot’s scientific life was fruitful and which one quantity depends on another, productive. He was a scientific adviser occurred in various contexts such as naviga- in the service of Sir Walter Raleigh, and a tion and cartography, theories of motion, pioneer of telescopic observation and of and celestial mechanics. The tools available early modern algebra. He contributed to deal with them, such as the theory of pro- to such diverse fields as linguistics, portions or diagrammatic representations of navigation, botany, astronomy, optics and change, mostly had their roots in antiquity mechanics, but published hardly any of his and had been further developed during the scientific findings. The treatise ‘Magisteria Middle Ages. Some of these tools, for Magna’, in which he presents his method example algebraic notation, were in rapid of interpolation, remained unpublished development. In the second half of the for almost 400 years; it has now been seventeenth century these developments edited by Janet Beery and Jacqueline led to the multiple invention of the calculus, Stedall. which was to change in a fundamental way Harriot’s exposition is almost non-verbal the method of dealing with functional but follows a clear and elegant line of relations. In the early seventeenth-century reasoning. He starts with a very basic manuscript edited in Thomas Harriot’s doc- table. The first row consists of ones. trine of triangular numbers, the English Below this is a row of sums of the ones mathematician and philosopher Thomas from the beginning to the place above the Harriot (1560–1621) develops an ingenious respective entry. The resulting sequence is tool for dealing with functional relations that of the natural numbers: 1, 2, 3, 4, .... that hitherto has been paid little attention The next row is again a row of sums in the history of mathematics: the use of of the entries of the previous row: 1, 3, 6,
303 This journal is q 2010 The Royal Society 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