Pocket calculators only came into common use in the 1970s. Digital computers first appeared in the 1940s, but were not in widespread use by the general public until the 1980s. Before pocket calculators, there were mechanical desktop calculators, but these could only add and subtract and at best do the most basic of multiplications.
A tool commonly used by engineers and scientists who dealt with math in their work was the slide rule. A slide rule is actually a simple form of what is called an analog computer, a device that does computation by measuring and operating on a continuously varying quantity, as opposed to the discreet units used by digital computers. While many analog computers work by measuring and adding voltages, the slide rule is based on adding distances.
A simple slide rule looks like two rulers that are attached to each other so that they can slide along one edge. In fact any two number lines that can be moved past each other make up a slide rule. To add two numbers on a slide rule, look at the above illustration, which shows how to add 3 and 4. Start by finding 3 on rule A. Then slide rule B so that the zero point of rule B lines up with 3 on rule A. Find 4 on rule B and it will be lined up with the sum on rule A, which is 7. Reverse these steps to subtract two numbers.
The slide rule was invented by William Oughtred (1574–1660), an English mathematician. Oughtred designed the common slide rule, which has a movable center rule between fixed upper and lower rules. Many different scales are printed on it, and a sliding cursor helps the user better view the alignment between scales. He also designed less common circular slide rules that worked on the same principles.
The previous example has already illustrated how to add and subtract using a slide rule. One can also multiply and divide with a slide rule using logarithms . In fact, slide rules were invented specifically for multiplying and dividing large numbers using logarithms shortly after John Napier invented logarithms.
To multiply 3,750 and 225 using a slide rule, first find log10 3,750 and log10 225. Use the slide rule to add these logarithms. Then find the antilog of the sum, which is the product of 3,750 and 225.
Division using a slide rule is the same as multiplication, except one subtracts the logarithms. Of course, one must have logarithm tables handy when multiplying or dividing with a slide rule, to look up the logarithms.
One can compute powers on a slide rule by taking the log of a log. Since A x is the same as x times log A, one can do this multiplication by adding log x and log (log A ). Special log-log scales on a slide rule make it possible to calculate powers. Roots, such as can be considered fractional powers, so the slide rule with a log-log scale could be used to compute roots as well. Additional scales were provided to look up trigonometric functions such as sine , cosine , and tangent .
The accuracy of a slide rule is limited by its size, the quality to which its scales are marked, and the ability of the user to read the scales. Typical engineering slide rules were accurate to within 0.1 percent. This degree of accuracy was considered sufficient for many engineering and scientific applications. However, the use of slide rules declined rapidly after the electronic calculator became inexpensive and widely used. This was due, in large part, to the fact that it is much easier and convenient to perform applications, such as multiplication, on a calculator than it is to look up logarithms.
see also Analog and Digital; Bases; Logarithms; Mathematical Devices, Early.
Harry J. Kuhman
Hopp, Peter M. Slide Rules: Their History, Models, and Makers. Mendham, NJ: Astragal Press, 1999.
Sommers, Hobart H., Harry Drell, and T. W. Wallschleger. The Slide Rule and How to Use It. New York: Follett, 1964.
The slide rule is an analog device for performing mathematical computations. The first slide rule was created in 1630 by British mathematician William Oughtred (1574–1660). His device was based on the logarithmic scale created by British astronomer Edmund Gunter (1581–1626) in 1620. Gunter's work, in turn, was based on the principle of logarithm set forth by Scottish mathematician John Napier (1550–1617) in 1614. What is known as the modern slide rule took shape during the first half of the 1800s.
The slide rule was used by scientists and engineers from 1640 through the 1960s and 1970s. The most familiar modern slide rule is basically a ruler with a sliding piece. Both parts of the device are marked with a scale of digits. The number of significant digits that a slide rule can contain is limited by the size of the rule. Circular and cylindrical slide rules were created to provide more significant digits. Circular slide rules operate on a set of concentric circles , while cylindrical slide rules use scales that can be manipulated by spinning them around a central rod. Today, slide rules are primarily a collector's item, having been replaced by computers and hand-held electronic calculators.
Logarithms and Slide Rules
In the 1600s, John Napier determined that any real number can be expressed as a power (log) of another number and these values could be published as logarithmic tables. He discovered that adding and subtracting logs is the same as multiplying and dividing real numbers.
The common logarithmic tables express real numbers as powers of ten. For example 2 is 100.3 and 2's log is 0.3. Also 2 2 2 8. 0.3 0.3 0.3 0.9, the log of 8. Edmund Gunter's logarithm scale arranged numbers from 1 to 10, spacing the numbers in proportion to their logs. Number 2 was spaced 0.3 from 1; 4 was 0.6 from 1, and so forth. Based on this scale, William Oughtred created the slide rule as an instrument for multiplying and dividing, based on logs. As mathematicians discovered new ways to use the slide rule, more scales were added to determine squares, roots, common logarithms, and trigonometry functions.
see also Analog Computing; Digital Computing; Napier's Bones.
Bertha Kugelman Morimoto
Thompson, J. E. The Standard Manual of the Slide Rule, 2nd ed., Princeton, NJ: D. Van Nostrand Co., Inc., 1952.
slide rule, instrument for making numerical computations and readings, the results of which may be read easily and quickly after performing simple mechanical manipulations. Multiplication and division, finding of powers and roots, and other more complicated calculations may be performed with a slide rule. Based on John Napier's principle of the logarithm, it came into use after Edmund Gunter created a logarithmic scale in 1620. Gunter's rule consisted of a straight line on which numbers were spaced at intervals proportional to their common logarithms. Using this scale, William Oughtred and Edmund Wingate developed independently (c.1630) the first slide rules. Amédée Mannheim, a French army officer, in 1850 established the form that it maintained thereafter. This had three parts, the stock, the slide, and the cursor (indicator). The stock consisted of two fixed parallel rules, each with a scale on its inner edge. The slide was a single rule, moving between them. It had two scales on its outer edge, each scale corresponding to the fixed scale to which it was adjacent. The cursor, a transparent square with a hairline, could be moved the length of the rule to aid in reading it. In its many varieties it had additional scales for such calculation as logarithms, trigonometric functions, and square roots. By the late 1980s they had been supplanted by the electronic calculators, which were easier to use and more accurate.
cur·sor / ˈkərsər/ • n. a movable indicator on a computer screen identifying the point that will be affected by input from the user, for example showing where typed text will be inserted. ∎ the transparent slide engraved with a hairline that is part of a slide rule.