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Problems concerning tangents and quadrature have a long history predating the Newton/Leibniz formulation of calculus; indeed, they are amongst the oldest problems in mathematics. It seems reasonable to assume that the relationship between the two problems was “well” understood for some time prior to the calculus.

According to Carl Boyer, in his A History of Mathematics, Isaac Barrow’s Lectiones geometriae (1670) prominently feature both problems, and it is “likely” that he understood the relationship between the two. However, Boyer also points out that Barrow’s dislike of analytic methods were “scarcely conducive to analytic discoveries” and that it was at Newton’s insistence that an analytic treatment of this material was included in Barrow’s text - Newton was acting as Barrow’s editor. Boyes also notes that Barrow’s conservative adherence to geometric methods kept him from making effective use of this relationship.

Many figures prior to Barrow appear to have been in position of all the necessary theory to positively identify the relationship - Fermat, Huygens, Wallis, Gregory, for example.

When was the inverse relationship between tangents and quadrature/area first identified?

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For this complex historical process, you can see:

See Introduction, page 4:

The infinitesimal calculus emerged in the form of two major classes of problem, the first concerned with the determination of arc, area, surface, volume and centre of gravity, the second with angle, chord, tangent, curvature, turning point and inflexion. [...] The inverse nature of the two classes of problem was approached in terms of geometric model by Torricelli, Gregory and Barrow but only with Newton did the relation emerge as central and general.

See:

[page 177] Fermat: for one of the earliest link between differential and integral processes;

[page 179] for Roberval relating tangent and area properties of curves;

[page 189] for Torricelli clear understanding (?) of the inverse relation between integration and differentiation;

[page 243] Isaac Barrow.

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  • $\begingroup$ It's interesting that Baron considers Barrow's approach to the relationship to be a geometric one, as was characteristic of his general approach to all his maths. On the other hand, Boyer notes that Newton insisted that Barrow treated this material in an analytic fashion. One assumes that Barrow's published treatment was therefore that of Newton rather than his own. $\endgroup$ – Nick Jun 9 '16 at 16:38

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