Barrow, Newton and Leibniz — a historical note

That differentiation and integration are inverse operations is the great discovery that, in the second half of the seventeenth century, turned a collection of techniques for areas and tangents into the infinitesimal calculus. Before then the problem of areas (the quadrature) and that of tangents were tackled by separate methods; the link between the two was first glimpsed in geometric form by Isaac Barrow, Newton’s teacher at Cambridge, in his Lectiones geometricae (1670): in a theorem now recognised as a geometric version of the fundamental theorem, Barrow showed that the tangent to the area curve has slope equal to the ordinate of the original curve.

It was Isaac Newton, however, in 1665–1666, who turned the result into a computational tool: in his method of fluxions the quantity that flows in time and its rate of change are linked precisely by the inversion between the calculus of areas and that of “fluxions”. Independently, in the 1670s, Gottfried Wilhelm Leibniz reached the same core with a notation destined to endure: the symbols $\int$ (an elongated S for summa) and $dx$ make it visible that

$\frac{d}{dx}\int_a^x f(t)\,dt = f(x),$

that is, that summing infinitesimal quantities and then differentiating returns the original function (Boyer). The priority of the discovery sparked a famous and bitter controversy between Newton’s supporters and Leibniz’s, lasting decades; today it is agreed that the two reached the calculus independently, and Leibniz’s notation — with $F(b)-F(a)$ for the definite integral — is the one we still use (Katz). Only much later, in the nineteenth century, did Cauchy and Riemann give the integral the rigorous definition that makes the theorem a genuine theorem rather than an operational principle (Stillwell).

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Topics: Calculus theorems
Concepts: Fundamental theorem of calculus · Antiderivative
People: Isaac Barrow · Isaac Newton · Gottfried Leibniz