A slide rule is a hand-operated mechanical calculator consisting of slidable rulers for conducting mathematical operations such as multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest analog computers.
Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance feature scales that aid in specialized calculations particular to those fields. The slide rule is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler, the slide rule is not meant to be used for measuring length or drawing straight lines. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results.
English mathematician and clergyman Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. It made calculations faster and less error-prone than evaluating on paper. Before the advent of the scientific pocket calculator, it was the most commonly used calculation tool in science and engineering. The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through the 1950s and 1960s even with the introduction of mainframe digital electronic computers. But after the handheld HP-35 scientific calculator was introduced in 1972 and became inexpensive in the mid-1970s, slide rules became largely obsolete and no longer were in use by the advent of personal desktop computers in the 1980s.
In the United States, the slide rule is colloquially called a slipstick.
Basic concepts Each ruler's scale has graduations labeled with precomputed outputs of various mathematical functions, acting as a lookup table that maps from position on the ruler as each function's input. Calculations that can be reduced to simple addition or subtraction using those precomputed functions can be solved by aligning the two rulers and reading the result.
For example, a number to be multiplied on one logarithmic-scale ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying the law of the logarithm of a product, the product of the two numbers can be read. More elaborate slide rules can perform other calculations, such as square roots, exponentials, and trigonometric functions.
The user may estimate the location of the decimal point in the result by mentally interpolating between labeled graduations. Scientific notation is used to track the decimal point for more precise calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.
Components
Most slide rules consist of three parts: Frame or base two strips of the same length held parallel to form a frame. Slide a center strip that can move lengthwise relative to the frame. Cursor, runner or glass an exterior sliding piece with a hairline for accurately reading and aligning numbers.
Some slide rules ("duplex" models) have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip (which can usually be pulled out, flipped over and reinserted for convenience), still others on one side only ("simplex" rules). A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales.
Decades Scales may be grouped in decades, where each decade corresponds to a range of numbers that spans a ratio of 10 (i.e. a range from 10n to 10n+1). For example, the range 1 to 10 is a single decade, and the range from 10 to 100 is another decade. Thus, single-decade scales (named C and D) range from 1 to 10 across the entire length of the slide rule, while double-decade scales (named A and B) range from 1 to 100 over the length of the slide rule.
Operation
Logarithmic scales The following logarithmic identities transform the operations of multiplication and division to addition and subtraction, respectively:
Multiplication With two logarithmic scales, the act of positioning the top scale to start at the bottom scale's label for corresponds to shifting the top logarithmic scale by a distance of . This aligns each top scale's number at offset with the bottom scale's number at position . Because , the mark on the bottom scale at that position corresponds to . With and for example, by positioning the top scale to start at the bottom scale's , the result of the multiplication can then be read on the bottom scale under the top scale's :
While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to is found by first positioning the top scale to start above the 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where is on the top scale:
But since the is above the second set of numbers that number must be multiplied by . Thus, even though the answer directly reads , the correct answer is .
For an example with even larger numbers, to multiply , the top scale is again positioned to start at the on the bottom scale. Since represents , all numbers in that scale are multiplied by . Thus, any answer in the second set of numbers is multiplied by . Since in the top scale represents , the answer must additionally be multiplied by . The answer directly reads . Multiply by and then by to get the actual answer: .
In general, the 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. This works because the distances from the mark are proportional to the logarithms of the marked values.
Division The illustration below demonstrates the computation of . The on the top scale is placed over the on the bottom scale. The resulting quotient, , can then be read below the top scale's :
There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the at either end.
With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus would be computed as and the result, , can be read beneath the in the top scale in the figure above, without the need to register the intermediate result for .
Solving proportions
Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems.
For example, setting 7.5 on one scale over 10 on the other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 5, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%.
Other scales
In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular are trigonometric, usually sine and tangent, common logarithm (log) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (ex) scales. 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. C, D single-decade logarithmic scales, single sections of the same length, used together for multiplication and division, and generally one of them is combined with another scale for other calculations A, B two-decade logarithmic scales, two sections each of which is half the length of the C and D scales, used for finding square roots and squares of numbers K three-decade logarithmic scale, three sections each of which is one third the length of the C and D scales, used for finding cube roots and cubes of numbers CF, DF folded versions of the C and D scales that start from pi (π) rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 and is not sure whether it will be slightly less or slightly more than 10, the folded scales avoid the possibility of going off the scale. Second, by making the start π rather than the square root of 10, multiplying or dividing by π (as is common in science and engineering formulas) is simplified. CI, DI, CIF, DIF inverted scales running from right to left, used to simplify reciprocal () steps S used for finding sines and cosines on the C (or D) scale T, T1, T2 used for finding tangents and cotangents on the C and CI (or D and DI) scales R1, R2 square root scales – setting the cursor to any value on R1 or R2, find (area of a circle of radius ) under the cursor on the DF scale ST, SRT used for sines and tangents of small angles and degree–radian conversion Sh, Sh1, Sh2 used for finding hyperbolic sines on the C (or D) scale Ch used for finding hyperbolic cosines on the C (or D) scale Th used for finding hyperbolic tangents on the C (or D) scale L linear scale used for addition, subtraction, and (along with the C and D scales) for finding base-10 logarithms and powers of 10 LL0N (or LL/N) and LLN log-log folded and scales, for working with logarithms of any base and arbitrary exponents. 4, 6, or 8 scales of this type are commonly seen. Ln linear scale used along with the C and D scales for finding natural (base ) logarithms and PPythagorean scale of to (1) solve the Pythagorean theorem and (2) to accurately determine cosine for small angles (with the S scale)
The scales on the front and back of a Keuffel and Esser (K&E) 4181-3 slide rule
Roots and powers There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute , 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 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 and square it using the A and B scales as described above. Alternatively, use the rightmost 1 on the C scale, and read the answer off the next higher LL scale. For example, aligning the rightmost 1 on the C scale with 2 on the LL2 scale, 3 on the C scale lines up with 8 on the LL3 scale.
To extract a cube root using a slide rule with only C/D and A/B scales, align 1 on the B cursor with the base number on the A scale (taking care as always to distinguish between the lower and upper halves of the A scale). Slide the slide until the number on the D scale which is against 1 on the C cursor is the same as the number on the B cursor which is against the base number on the A scale. (Examples: A 8, B 2, C 1, D 2; A 27, B 3, C 1, D 3.)
Roots of quadratic equations