Graduate studies at Western
|Abstract||As is well known, one of Leibniz’s seminal insights in his work on series concerned sums of differences. If from a given series A one forms a difference series B whose terms are the differences of the successive terms of A, the sum of the terms in the B series is simply the difference between the last and first terms of the original series: “the sum of the differences is the difference between the first term and the last” (A vii.3, p. 95). This insight, so far as I now, has never been named; I shall call it the Difference Principle. Suitably generalized, it becomes the basis for the fundamental theorem of the calculus: the sum (integral) of the differentials equals the difference of the sums (the definite integral evaluated between last and first terms), ∫ Bdx = [A]fi. As is well known, the Difference Principle has its origin in the problem set him by Huygens in September 1672 to find the sum of the reciprocal triangular numbers. This date is, incidentally, confirmed by Leibniz himself in Summa fractionum a figuratis, per aequationes (A vii.3, p. 365 ), as well as by the first piece in A vii.3, De summa numerorum triangulorum recipricorum (p. 3), although in his Origo inventionis trianguli harmonici of the Winter of 1675-76 he misremembers it as “Anno 1673” (p. 712). Leibniz explains the algorithm in a letter to Meissner 21 years later: “If one wants to add, for example, the first five..|
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