Bernhard Riemann (1826-1866) was a German mathematician. Riemann made
routine use of infinitesimal techniques. Thus, in his famous 1854
essay *On the Hypotheses which lie at the Foundations of
Geometry*, Riemann speaks of the line as being made up of the
*dx*, describes *dx* as `the increments', and speaks of
infinitesimal displacements and of infinitely small quantities
*x _{1} dx_{2} - x_{2} dx_{1}*,
etc., as well as of infinitely small triangles. In talking about the
`next-order term' in the expansion of the line element on a manifold,
Riemann writes:

It obviously equal to zero if the manifold in question is flat, i.e., if the square of the line element is reducible to

It is apparent from this passage concerning the Gaussian curvature that Riemann has a pointwise (local) discussion of

What meaning did Riemann assign to dx? (MSE)

Fermat

Leibniz

Euler

Cauchy

Cantor

Robinson

Nelson

Hrbacek

Infinitesimal topics

More on infinitesimals

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