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He stated that the third proportional between ndz = d−1 (ndz) and ndz = d0 (ndz) was d(ndz). Indeed, if one operated in the same way as in Leibniz’s analogy, one had d−1 (ndz) : d0 (ndz) = d0 (ndz) : d(ndz). By diﬀerentiating d(ndz), he obtained d(ndz) = d0 nd2 z + dndz and, therefore, ndz = d−1 (ndz) = d0 nd2 z d0 ndz (d0 (ndz))2 = 0 2 = 0 . d(ndz) d nd z + dndz d ndz + dn 67 In 1710 Leibniz published this analogy in a paper entitled Symbolismus memorabilis calculi algebraici et inﬁnitesimalis in comparatione potentiarum et diﬀerentiarum, et de lege homogeneorum transendentali .
By dividing such a remainder by 1 + x, one obtains the quotient x and the remainder −x2 . By continuing in inﬁnitum, one obtains the series 1 − x + x2 − . .. ∗ ∗ ∗ The last mathematician I shall discuss in this chapter is James Gregory. He made a number of remarkable contributions to series theory and some of his results overlapped with the ﬁndings of Newton. In his Vera Circuli et Hyperbolae Quadratura , while investigating the areas of conic sections, he introduced the expression “convergent series” [1667, 10].
AM AB + BM By expanding, he had M L = AB − BM + BM 2 BM 3 BM 4 BM 5 − + − + ... AB AB 2 AB 3 AB 4 under the condition BM < AB. Leibniz applied some theorems that he had proved earlier in De quadratura arithmetica and showed that the area of the ﬁgure BEHC (= to the sum all M L from BC to EH) was given from n the sum of the areas of the curves having ordinate equal to BM AB m (with M varying from B to E). Since AB is constant, the area AB is AB · AB; the 2 area of BM is a triangle of sides BE and BE, namely BE 2 , and so on.