Special Scales

We have already seen how most slide rules, in addition to the basic C and D scales used for multiplication, have the A and B scales, useful for problems involving squares and square roots, the K scale for cubes and cube roots, the CI scale for reciprocals, and the L scale on which the common logarithms of numbers on the C or D scale can be read.

The Trigonometric Scales

Still more scales were very common on slide rules. Multiplying numbers by the sine, cosine, or tangent of an angle is something that one doesn't need to be a rocket scientist to have to do on occasion; even being a carpenter will suffice, and, therefore, trigonometric scales were, at least during the 20th Century, a standard feature even on most basic slide rules.

The S scale has marks that are labelled with angles in degrees; the location of each mark corresponds to the position on the C scale where the sine of that angle is found. Similarly, the T scale has marks that are labelled with angles in degrees, and the location of each mark corresponds to the position on the C scale where the tangent of that angle is found.

Thus, the S and T scales can be used for multiplying a number by the sine or tangent of an angle. And, of course, the cosine of 90 degrees minus an angle is the sine of that angle.

The tangent of an angle over 45 degrees is the reciprocal of the tangent of 90 degrees minus that angle. Thus, one uses the T scale to divide instead of multiply for such large angles. The number of degrees for such angles is thus noted in red on most slide rules, in the same way that the numbers on the CI scale usually are.

Usually, the center slide of even a simple slide rule had S and T scales. The simplest way to work with those scales is by flipping the center part over by taking it out of the side of the slide rule first, then flipping it over and putting it back in. But there were windows in the back of the slide rule that permitted using those scales without doing this, although one had to remember how to do that. The L scale was usually between the S and T scales on the back side of the center slide. On these slide rules, the S and T scales tended to be marked in degrees and minutes, and the S scale might work with the A and B scales while the T scale worked with the C and D scales.

The Log-Log Scales

The fancier slide rules also had a set of scales called "log log scales". What these scales let you do was raise numbers to arbitrary powers.

If the K scale, for example, shows the cube of numbers on the D scale by including three scales in the space of one, because adding the logarithm of a number to itself three times produces the logarithm of its cube, then one can cube a number by multiplying its logarithm by three.

And in that case, since a slide rule performs multiplication, a suitable scale should allow exponentiation.

This diagram illustrates a slide rule with a very small-scale log-log scale, matching the K scale. Usually, the log-log scale shown here would instead be broken into three pieces, called LL1, LL2, and LL3, matching the C scale.

The coarse resolution of the diagram above, and its small scale, shows the mark for 1.001 on the LL scale coinciding with the 1 on the K scale, but when the scale is made to correspond with the C scale, on a real 10 inch slide rule, a fourth piece, extending from 1.0001 to 1.001, called the LL0 scale, is worthwhile to provide, even if, for smaller numbers, the D scale will serve.

By shrinking the log-log scale down, I paired a scale with both it and the K scale in the diagram, to make the illustration simpler to understand.

The J scale has been moved so that the 1 on that scale corresponds to the 3 on the K scale. Thus, every number on the J scale is multiplied by three on the K scale.

This displacement means that every number on the MM scale is cubed on the LL scale.

Thus, 1.01 is just past 1.03; 2 is under 8, 3 is under 27, 10 is under 1000.

Notice that the distance between numbers quickly shrinks. Thus, the tick marks between 100 and 1000 represent 150, 200, 250, 300, 350, 400, and 500.

While 10, 100, and 1000 stand in the same relation to each other on the LL scale as 1, 2, and 3 do on the K scale, note that 1.01, 1.02, and 1.03 also stand in approximately the same relationship