The beginnings of integration can be recognised in the work of the ancient Greeks (Euclid, Archimedes ) in finding areas of curved regions and volumes of curved solids.

The beginnings of differentiation were much later, in the work of the early 17th century on tangents to curves and instantaneous rates of change.

The recognition that these two processes are inverses of each other (the "Fundamental Theorem of Calculus") and the major initial development of the theory occurred in the late 17th century, mainly in the work of Newton (1642-1727) and Leibniz (1646-1716).

All calculus was based on the concept of a limit, a concept which was not well understood until the 19th century (in the work of Cauchy, Riemann, Weierstrass and others) and until then the results in the calculus were founded on an unsound, non- rigorous basis.

(e.g. one intuitive idea was that the gradient of the tangent

to the curve at (x,y) is the gradient of the chord, i.e.

when x = 0.

For instance if y = x, then gradient of chord = = = 2x + x

- and when x = 0 this is 2x.

But when x = 0 the chord does not exist! )

The Beginnings of Integration

The Development of Calculus

History of Mathematics Module

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These pages are maintained by M.I.Woodcock.