I spent a good portion of the day learning about slide rules because I received my Concise No. 28 Circular Slide rule yesterday and I want to start using it for calculations. I believe that it encourages good calculation habits because it demands efficient calculations. I looked for reference materials by searching through manuals at http://sliderule.ozmanor.com/man/man-download.html and I found that the Keuffel & Esser manual had the best information on the theory of slide rules. It quickly allowed me to grasp the ideas behind a slide rule and therefore its operation.
NB: I only understood the manual because I understand the properties of logarithms and exponents. Please read up on that before you try to understand the following breakdown of what I learned.
The section on slide rule theory starts by comparing logarithmic slide rule operation to linear slide ruler operation. With the linear slide ruler, one can perform addition and subtraction by setting units together side by side. Since we can perform addition and subtraction with rulers, we can do something more interesting -- add and subtract scaled units. It turns out that using logarithmic scaled units performs multiplication and division.
Most operations can be done with plain logarithmic scales. I won't go into the plain scales, but I will go into a cool variant called the "folded scale," which typically starts at Pi, but could start at any number. If the folded scale starts at Pi (in other words, if the index is Pi), when the slide is closed, Pi is above "1" on the regular scale and the indexes are lined up. So, by moving the cursor to any position on the regular scale, the number it lands on will be the multiplier of Pi. The product of Pi and this number will be under the cursor on the folded scale. The folded scale also goes through a change in distance equal to the change you make on the unfolded scale. Because of this, products that fall off the end of a linear slide rule scale can usually be picked up on the folded scale and read off. On my circular slide rule, I don't need to worry about ever falling off the edge because a circle has no edges.
The inverse scale is a variant of the original scale. Simply, it is the original logarithmic scale laid down from right to left instead of from left to right. It is laid down like this because the entire length of the original scale is log(10). When you set the cursor, you break it into two pieces. One piece is length log(N) and the other length log(10) - log(N) = log(10/N). Since log(10/N) is the same as the inverse with a different characteristic, if you measure it, you are measuring log(1/N). So, the cursor would be on 1/N if you lay a scale down from the right.
The greatest power of a slide rule is not in doing single multiplication and division problems. It's greatest power is to view scale (K x) and proportion (a/b = c/d) relationships at a glance. By multiplying by a number, K, you can find all products of K by setting the cursor to your desired multiplicand. By setting two numbers on top of each other, you are in effect setting all other marks between the scales to reflect that proportion. To find an equivalent proportion, simply set the cursor. It's remarkable that we don't use slide rules still because of this amazing property.
After writing this, I realized that the slide rule is an example of the broader topic of transformations. Transformations in general are used to make operations easier or faster to compute. The most famous in electrical engineering are the Laplace, Fourier, and Phasor transforms. Where logarithms transform the operation of multiplication to the easier operation of addition, these transformations change functions to make differential equations easier to solve.
