Moving the top scale to the right by a distance of formula_3, by matching the beginning of the top scale with the label formula_4 on the bottom, aligns each number formula_5, at position formula_6 on the top scale, with the number at position formula_7 on the bottom scale. Because formula_8, this position on the bottom scale gives formula_9, the product of formula_4 and formula_5. For example, to calculate 3*2, the 1 on the top scale is moved to the 2 on the bottom scale. The answer, 6, is read off the bottom scale where 3 is on the top scale. In general, the 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top.
Operations may go "off the scale;" for example, the diagram above shows that the slide rule has not positioned the 7 on the upper scale above any number on the lower scale, so it does not give any answer for 2×7. In such cases, the user may slide the upper scale to the left until its right index aligns with the 2, effectively multiplying by 0.2 instead of by 2, as in the illustration below:
Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer. We wanted to find 2×7, but instead we calculated 0.2×7=1.4. So the true answer is not 1.4 but 14. Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as 2×7; some other methods are:
Method 1 is easy to understand, but entails a loss of precision. Method 3 has the advantage that it only involves two scales.
The illustration below demonstrates the computation of 5.5/2. The 2 on the top scale is placed over the 5.5 on the bottom scale. The 1 on the top scale lies above the quotient, 2.75. There is more than one method for doing division, but the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end.
In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular were trigonometric, usually sine and tangent, common logarithm (log) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential ("e") scales. Some rules include a Pythagorean scale, to figure sides of triangles, and a scale to figure circles. Others feature scales for calculating hyperbolic functions. On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order:
The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions.
There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute formula_12, for example, locate x on the D scale and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90.
For formula_13 problems, use the LL scales. When several LL scales are present, use the one with x on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find y on the C scale and go down to the LL scale with x on it. That scale will indicate the answer. If y is "off the scale," locate formula_14 and square it using the A and B scales as described above.
The S, T, and ST scales are used for trigonometric functions and multiples of trig functions, for angles in degrees. Many slide rules have their S, T, and ST scales marked with degrees and minutes. So-called "decitrig" models use decimal fractions of degrees instead.
Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which is for base e.
The Ln scale was invented by an 11th grade student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent "x" (in the range 0 to 2.3) on the Ln scale and read e on the C (or D) scale and e on the CI (or DI) scale. Pickett, Inc. was given exclusive rights to the scale. Later, the inventor created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of its slide rules.
Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques.
The first method to perform addition and subtraction on the C and D (or any comparable scales) requires converting the problem into one of division. For addition, the quotient of the two variables plus one times the divisor equals their sum:
For subtraction, the quotient of the two variables minus one times the divisor equals their difference:
This method is similar to the addition/subtraction technique used for high-speed electronic circuits with the logarithmic number system in specialized computer applications like the Gravity Pipe (GRAPE) supercomputer and hidden Markov models.
The second method uses a sliding linear L scale available on some models. Addition and subtraction are performed by sliding the cursor left (for subtraction) or right (for addition) then returning the slide to 0 to read the result.
The length of the slide rule is quoted in terms of the nominal length of the scales. Scales on the most common "10-inch" models are actually 25 cm in length, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflowed. Pocket rules are typically 5 inches. Models a couple of meters long were sold to be hung in classrooms for teaching purposes.
Typically the divisions mark a scale to a precision of two significant figures, and the user estimates the third figure. Some high-end slide rules have magnifying cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10-inch slide rule to serve as well as a 20-inch.
Various other conveniences have been developed. Trigonometric scales are sometimes dual-labeled, in black and red, with complementary angles, the so-called "Darmstadt" style. Duplex slide rules often duplicate some of the scales on the back. Scales are often "split" to get higher accuracy.
Specialized slide rules were invented for various forms of engineering, business and banking. These often had common calculations directly expressed as special scales, for example loan calculations, optimal purchase quantities, or particular engineering equations. For example, the Fisher Controls company distributed a customized slide rule adapted to solving the equations used for selecting the proper size of industrial flow control valves.
Circular slide rules come in two basic types, one with two cursors (left), and another with a movable disk and a single cursor (right). The dual cursor versions perform multiplication and division by maintaining a fixed angle between the cursors as they are rotated around the dial. The single cursor version operates more like the standard slide rule through the appropriate alignment of the scales.
The basic advantage of a circular slide rule is that the longest dimension of the tool was reduced by a factor of about 3 (i.e. by π). For example, the outer scale of a 10 cm circular would have a maximum precision equal to a 30 cm ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be re-oriented when results are near 1.0—the rule is always on scale. However, for non-cyclical non-spiral scales such as S, T, and LL's, the scale length is shortened to make room for end margins.