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Slide Rules with spiral scales are almost exclusively based on Archimedian spirals. Here are two spiral scales, the first use the customary Archimedian layout; the second uses a Log Spiral layout. Both use the same inner and outer radius
Archimedian Spiral
Archimedian Spiral
Every tick "i" along the Archimedian spiral is located at at an angle proportional to log (i). The radial distance is linearly proportional to the angle.
The spiral shape means that the tick spaces are further apart on the outer edge of the spiral than they would be on a linear scale of the same length. However, the tick s are not equidistant around the spiral coils.
Logarithmic Spiral
Logarithmic Spiral
Every tick "i" along the Logarithmic Spiral is located along the line at at an distance that is linearly proportional to i. The radial distance is exponentially proportional to the angle. It is easily seen that the coils are more closely spaced at the center than at the outer edge. Also you may be able to see that each tick is exactly the same distance apart. If you could unravel the spiral into a straight line, it would become a common ruler.
This linearity of tick spacing was a key feature exploited by George MacDonald in this invention. He added verniers to the to the spiral scale to increase the precision by up to 2 digits.
The full derivation for the Log spiral slide rule is in the patent description.
R = Ri * e^(a * theta) and a = LOGe(10) / 2 * Pi * Turns
Where
R is the radial distance
Ri = is the inner diameter
Theta = 1/a * q * sqrt (a^2 + 1) * n
Where
q is the relative tick spacing, or precision. (.001) means "1 in 1,000", .0001 means "1 in 10,000"
n is the 'nth tick'. E.g., " 0 - 900" for "1 in 1,000 precision", or "0 - 9000" for "1 in 10,000 precision"
Not that "q" and "a" are constants, so theta is linearly proportional to n
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