Symbol d

The passage around this formula

Summation is the easy operation to build. Integration is the one that opens up a whole class of problems, and the mechanism that made it practical was the wheel-and-disc integrator: a wheel riding on a rotating disc at a radius that can be varied, so that the wheel’s rotation accumulates the product of the disc’s rotation and the instantaneous radius. If the disc rotation represents the independent variable and the radial position represents the dependent one, the wheel angle represents $z(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau$. which is to say, the integral of y with respect to x , produced continuously and without arithmetic.

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