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By Janet Beery and Jacqueline Stedall, Janet L. Beery, Jacqueline A. Stedall

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As an example he offered the equation aa C 6a D ‘a number’, and gave the tables from (W4) (Tables 30 and 31), demonstrating 9 interpolations between the values 7 and 16. Mercator went on to give Warner’s other examples of polynomial values, with the tables for aaa C 2aa C 3a D 1230 (see Table 29), and for aaaa C 1aaa C 3aa C 2a D ‘a number’, both from (W4). He also gave the rules for constructing the crucial first row: 1. First of the squares The first is the sum of the coefficients The second is the same The third is the Characteristick 2 2.

350–372 and 4396, ff. 20–29. 65 By the late 1620s Warner’s handwriting had become very shaky, see, for example, some draft material for the Praxis in BL Add MS 4395, f. 92. 63 30 Thomas Harriot’s ‘Magisteria magna’ (W3) A sheet beginning ‘Methodus calculatoria, qua canon analogicus …’ Now BL Add MS 4396, f. 19 (with associated rough working on f. 18). (W4) A method for solving polynomial equations using difference tables. Now BL Add MS 4395, ff. 166–166v and 181. (W5) A list of polynomials, evaluated for the first few positive integers, with successive differences.

109 Wallis 1656, Proposition 169. 108 48 Thomas Harriot’s ‘Magisteria magna’ 1 1 1 1 1 1 : : 1 1 1 1 1 ::: 2 3 4 5 6 ::: 3 6 10 15 21 : : : 4 10 20 35 56 : : : 5 15 35 70 126 : : : 6 21 56 126 252 : : : : : : : : : : : : : Table 37. Wallis’s table of triangular numbers.

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