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Equation 10 · Part 13 · The Machine Before Electronics

Numerator: dx

z(t)=∫0ty(τ) dxdτ dτz(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau
dxdx

What this part means

The complete quantity above the fraction bar.

Its job in the formula

dx occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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)=∫0ty(τ) dxdτ dτz(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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.