# Differentiation and integration

Which came first : Differentiation or Integration? If one of them was developed to solve certain types of problems, was the other developed for backward compatibility, or was it an independent development and later discovered that they were inverses?

Also, how were they discovered to be linked?

It was just something that I was thinking, and I would like to clarify it.

(I see that this is related : History of differential and integral calculus, but no answer is present)

• Answering questions like this comes down to what you mean exactly by differentiation and integration. As stated in the question you linked (and elsewhere) the ancient Greeks and Romans used techniques like the method of exhaustion. Is this a proto-integration technique? Is it a proto-limits technique? If you are meaning our modern form linked to functions then you'll get a different date/answer. – Ian Miller Mar 20 '16 at 5:58
• Not an answer, but I do know that they developed more or less independently of each other. For a long time, the connection between the two was not obvious, whereas today they are always taught as being intimately related by the fundamental theorem. Integration developed (at least in part) with the very geometric goal of finding areas under curves. – Elliot G Mar 20 '16 at 5:58
• Check out the website below. It seems like the short answer is that integration came first (in the form of areas under curves) and differentiation later (in the form of tangent lines to curves). math.ucdavis.edu/~temple/MAT16A/ArticlesOnCalculus16A/… – Elliot G Mar 20 '16 at 6:01

In a nutshell, the calculus was "discovered" or "invented" during 17th century independently by Leibniz and Newton who merged brilliantly various techniques developed since ancient Greece to solve geometrical problems.

Following the development of algebra during the Reanaissance and the pubblication of Descartes' Geometry in 1637, those methods were improved and new ones were discovered:

• drawing the normal to a curve: Descartes, Hudde

• finding tangents: Roberval, Fermat

• finding maxima and minima of curves: Fermat

• the method of indivisibles: Cvalieri

• arithmetical methods of integration: Wallis.

See:

The "official" birth of the calculus must be dated with Newton (De analysi of 1669) and Leibniz (various Ms. of 1675) independent developments:

Leibniz's differential and integral calculus and Newton's fluxional calculus, though different in many aspects, each involve a clear recognition of what we now call the inverse relationship between differentiation and integration. Moreover, both men worked out a system of notations, symbols and rules through which their methods could be applied in the form of algorithms performed on formulae, rather than in the form of geometrical arguments presented in prose with reference to figures.

See into: Ivor Grattan-Guinness, cit., Ch.2 Newton, Leibniz and the Leibnizian Tradition, by H.J.M. Bos, page 49-on

This is similar to the chicken-and-the-egg question though relation is not as obvious in the case of integration-and-differentiation. Namely, both types of problems were already studied by Archimedes (areas of figures and volumes of solids), Apollonius (tangents to curves), and others thousands of years ago. However the inverse relationship between the integration and differentiation was not understood until Barrow (some say even earlier by James Gregory), Newton, and Leibniz.