**Slide rules** were the **analog computers** that ruled science and engineering for 400 years. Their brilliant innovation was using **logarithms** to convert multiplication and division to addition and subtraction,

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\log \frac{x}{y} = \log x - \log y.Slide rules feature logarithmic scales that slide past each other. For **straight** slide rules, logarithms of the numbers are proportional to their distances along them. To multiply 2\times 3 = 6, as below, slide the upper (blue) scale from 1 to 2 along the lower (red) scale, add the distance from 1 to 3 along the upper (blue) scale, and read the product from the lower (red) scale.

For **circular** slide rules, logarithms of the numbers are proportional to their distances around them. To multiply 3\times 7 = 21, as below, rotate the outer (blue) scale from 1 to 3 around the inner (red) scale, add the distance from 1 to 7 around the outer (blue) scale, and read the product from the inner (red) scale. Circular slide rules eliminate **off-scale** calculations because they naturally **wrap around**, with each wrap multiplying (or dividing) the numbers by 10.