The problem with this, is that infinite series were in rudiment already known by Archimedes (his method of exhaustion), and that neither Newton nor Leibniz relied on limiting arguments in the development of calculus. Calculus was developed on the basis of infinitesimals or fluxions, which (with all the properties needed) are incompatible with classical logic (cf. a tract by Bp. Berekely (C of E) attacking Edmund Halley).
Limiting arguments, which tie the infinitesimal calculus to infinite series were developed about 150 years later by Cauchy.
Find any use of inifinitesimals before Newton and Leibniz and you have a case.
The nearest thing one has (but only seen through the lens of Descarte’s coordinate geometry) is the assertion of the sophist Heraclites, in contradiction of one of Euclid’s postulates, that the intersection between a circle and a tangent line is larger than a point. (From the point of view of infinitesimal calculus, while it contains only one point, it also contains all (first order) infinitesimal deformations of the point in the tangent direction.)
(My doctoral dissertation had to do with models in the context of intuitionistic logic of the sort of infinitesimals Newton and Leibniz actually used. When I teach calculus, I tell the students they have a choice; they can learn limits or substitute a graduate course in mathematical logic, thereby putting the original way of doing calculus on a sound footing.)
Excellent post!
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