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As we just saw, Descartes renovated mathematics in the early seventeenth century by bringing geometry and the algebra of arithmetic into novel combination. The middle term, setting number and figure into unprecedented relation, was the kind of algebraic equation we remember from high school: a few variables, a few constants, positive whole number exponents, a finite number of terms: something like ax² + by² = c. Descartes was quite finitist: he circumscribed his domain of research by limiting geometry to certain specified means of construction and representation. As the seventeenth century unfolded, however, this development brought about by the shifting conjunctions of number and figure created a demand for new notation, new middles terms: the infinite series, negative and fractional exponents, expressions for infinitesimal magnitudes and for new kinds of (infinitary) sums and for differential equations. Aristotle's restriction of the rational to the finite was challenged, so that finitude (as Pascal complained) suddenly found itself located between the infinitely small and the infinitely large. Moreover, the notion of geometric figure metamorphosed into that of a more generalized curve (some algebraic but some transcendental) and finally to the very general notion of a function; and the structure of space itself became an object of interest. Number expanded from the natural numbers, to the integers, to the rational numbers, finally gesturing towards the reals, number somehow endowed with the continuity of a line. The development of the infinitesimal calculus opened a large new chapter in the history of mathematics. Although techniques of integration date back to the time of Archimedes, as we noted earlier, the infinitesimal calculus properly began with the work of Newton and Leibnitz, marked by the conscious introduction and exploration of transcendental curves, algorithms for finding derivatives and integrals, and the formulation of the Fundamental Theorem of the Calculus, which exhibits the duality of integration and differentiation.

DOI: 10.1007/978-3-319-98231-1_10

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