The slide rule, also known colloquially in the United States as a slipstick,[1][2] is a mechanical analog computer.[3][4][5][6][7] As graphical analog calculators, slide rules are closely related to nomograms, but the former are used for general calculations, whereas the latter are used for application-specific computations.
The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms, and trigonometry, but typically not for addition or subtraction. 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.
Slide rules exist in a diverse range of styles and generally appear in a linear or circular form with a standardized set of graduated markings (scales) essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations particular to those fields.
At its simplest, each number to be multiplied is represented by a length on a sliding ruler. As the rulers each have a logarithmic scale, it is possible to align them to read the sum of the logarithms, and hence calculate the product of the two numbers.
The Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the electronic calculator, it was the most commonly used calculation tool in science and engineering.[8] The use of slide rules continued to grow through the 1950s and 1960s even as computers were being gradually introduced; but around 1974 the handheld electronic scientific calculator made them largely obsolete[9][10][11][12] and most suppliers left the business.
Slide rules exist in a diverse range of styles and generally appear in a linear or circular form with a standardized set of graduated markings (scales) essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations particular to those fields.
At its simplest, each number to be multiplied is represented by a length on a sliding ruler. As the rulers each have a logarithmic scale, it is possible to align them to read the sum of the logarithms, and hence calculate the product of the two numbers.
The Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the electronic calculator, it was the most commonly used calculation tool in science and engineering.[8] The use of slide rules continued to grow through the 1950s and 1960s even as computers were being gradually introduced; but around 1974 the handheld electronic scientific calculator made them largely obsolete[9][10][11][12] and most suppliers left the business.
In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers. These common operations can be time-consuming and error-prone when done on paper. More elaborate slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.
Scales may be grouped in decades, which are numbers ranging from 1 to 10 (i.e. 10n to 10n+1). Thus single decade scales C and D range from 1 to 10 across the entire width of the slide rule while double decade scales A and B range from 1 to 100 over the width of the slide rule.
In general, mathematical calculations are performed by aligning a mark on the sliding central strip with a mark on one of the fixed strips, and then observing the relative positions of other marks on the strips. Numbers aligned with the marks give the approximate value of the product, quotient, or other calculated result.
The user determines the location of the decimal point in the result, based on mental estimation. Scientific notation is used to track the decimal point in more formal calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.
Most slide rules consist of three parts:
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.
A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules and and . Moving the top scale to the right by a distance of , by matching the beginning of the top scale with the label on the bottom, aligns each number , at position on the top scale, with the number at position on the bottom scale. Because , this position on the bottom scale gives , the product of and . 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. This works because the distances from the "1" are proportional to the logarithms of the marked values:
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 dividing by 10 (by subtracting the full length of the C-scale) and then multiplying by 7, as in the illustration below:
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 dividing by 10 (by subtracting the full length of the C-scale) and then multiplying by 7, 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 (2/10)×7=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
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 are trigonometric, usually sine and tangent, common logarithm (log10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (ex) scales. Some rules include a Pythagorean ("P") 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: