The slide rule, also known colloquially as a slipstick,^{[1]} is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for "scientific" functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.
Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of 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 common to that field.
William Oughtred and others developed the slide rule in the 1600s based on the emerging work on logarithms by John Napier. Before the advent of the pocket calculator, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were being gradually introduced; but around 1974 the electronic scientific calculator made it largely obsolete and most suppliers exited the business.
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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 timeconsuming and errorprone when done on paper. More complex slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.
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 linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change.
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 could 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 log(xy) = log(x) + log(y) and log(x / y) = log(x) − log(y). Moving the top scale to the right by a distance of log(x), by matching the beginning of the top scale with the label x on the bottom, aligns each number y, at position log(y) on the top scale, with the number at position log(x) + log(y) on the bottom scale. Because log(x) + log(y) = log(xy), this position on the bottom scale gives xy, the product of x and y. 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 offscale 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 offscale, 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 (log10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (e^{x}) 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:
A, B  twodecade logarithmic scales, used for finding square roots and squares of numbers 
C, D  singledecade logarithmic scales 
K  threedecade logarithmic scale, used for finding cube roots and cubes of numbers 
CF, DF  "folded" versions of the C and D scales that start from π rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 but isn't 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, DIF  "inverted" scales, running from right to left, used to simplify 1/x steps 
S  used for finding sines and cosines on the D scale 
T  used for finding tangents and cotangents on the D and DI scales 
ST, SRT  used for sines and tangents of small angles and degree–radian conversion 
L  a linear scale, used along with the C and D scales for finding base10 logarithms and powers of 10 
LLn  a set of loglog scales, used for finding logarithms and exponentials of numbers 
Ln  a linear scale, used along with the C and D scales for finding natural (base e) logarithms and e^{x} 


The scales on the front and back of a K&E 40813 slide rule. 
The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions.^{[2]}
There are singledecade (C and D), doubledecade (A and B), and tripledecade (K) scales. To compute x^{2}, 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 x^{y} 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 x^{y / 2} and square it using the A and B scales as described above.
The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.
For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C. The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with C or, for angles greater than 45 degrees, CI. Common forms such as ksinx can be read directly from x on the S scale to the result on the D scale, when the Cscale index is set at k. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/radian. Inverse trigonometric functions are found by reversing the process.
Many slide rules have S, T, and ST scales marked with degrees and minutes. Socalled decitrig models use decimal fractions of degrees instead.
Base10 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^{x} on the C (or D) scale and e^{–x} 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.^{[citation needed]}
Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques.^{[3]}
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 highspeed electronic circuits with the logarithmic number system in specialized computer applications like the Gravity Pipe (GRAPE) supercomputer and hidden Markov models.
The second method utilizes 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 "10inch" 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. ^{[4]}
Typically the divisions mark a scale to a precision of two significant figures, and the user estimates the third figure. Some highend slide rules have magnifying cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10inch slide rule to serve as well as a 20inch.
Various other conveniences have been developed. Trigonometric scales are sometimes duallabeled, in black and red, with complementary angles, the socalled "Darmstadt" style. Duplex slide rules often duplicate some of the scales on the back. Scales are often "split" to get higher accuracy.
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, a 10 cm circular would have a maximum precision equal to a 30 cm ordinary slide rule. Circular slide rules also eliminate "offscale" calculations, because the scales were designed to "wrap around"; they never have to be reoriented when results are near 1.0—the rule is always on scale. However, for noncyclical nonspiral scales such as S, T, and LL's, the scale length is shortened to make room for end margins.^{[5]}
Circular slide rules are mechanically more rugged and smoothermoving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm offcentre of the pivot can result in a 0.2 mm worst case alignment error. The pivot, however, does prevent scratching of the face and cursors. The highest accuracy scales are placed on the outer rings. Rather than "split" scales, highend circular rules use spiral scales for more complex operations like logoflog scales. One eightinch premium circular rule had a 50inch spiral loglog scale.
The main disadvantages of circular slide rules are the difficulty in locating figures along a rotating disc, and limited number of scales. Another drawback of circular slide rules is that lessimportant scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch.
One slide rule remaining in daily use around the world is the E6B. This is a circular slide rule first created in the 1930s for aircraft pilots to help with dead reckoning. With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The socalled "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions, the E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of proficiency in its use.
Proportion wheels are simple circular slide rules used in graphic design to enlarge or reduce images and photographs. Lining up the desired values on the outer and inner wheels (which correspond to the original and desired sizes) will display the proportion as a percentage in a small window. They are not as common since the advent of computerized layout, but are still made and used.
In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer–nautical mile and gallon–liter fuel amount conversion functions.
There are two main types of cylindrical slide rules: those with helical scales such as the Fuller, the Otis King and the Bygrave slide rule, and those with bars, such as the Thacher and some Loga models. In either case, the advantage is a much longer scale, and hence potentially higher accuracy, than a straight or circular rule.
Traditionally slide rules were made out of hard wood such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument was made of steel.
In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable, strong and naturally selflubricating. These bamboo slide rules were introduced in Sweden in September, 1933,^{[6]} and probably only a little earlier in Germany. Scales were made of celluloid or plastic. Later slide rules were made of plastic, or aluminium painted with plastic. Later cursors were acrylics or polycarbonates sliding on Teflon bearings.
All premium slide rules had numbers and scales engraved, and then filled with paint or other resin. Painted or imprinted slide rules were viewed as inferior because the markings could wear off. Nevertheless, Pickett, probably America's most successful slide rule company, made all printed scales. Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. The recommended cleaning method for engraved markings is to scrub lightly with steelwool. For painted slide rules, and the faint of heart, use diluted commercial windowcleaning fluid and a soft cloth.
The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593–1656), an English mathematician, in a book entitled L'usage de la reigle de proportion en l'arithmetique & geometrie. The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 he combined two Gunter rules, held together with the hands, to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students, but delayed in publishing them, and like Newton, he became involved in a vitriolic controversy over priority, with his onetime student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.
In 1677, Henry Coggeshall created a twofoot folding rule for timber measure, called the Coggeshall slide rule. His design and uses for the tool gave the slide rule purpose outside of mathematical inquiry.
In 1722, Warner introduced the two and threedecade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.
In 1815, Peter Mark Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.
The more modern form was created in 1859 by French artillery lieutenant Amédée Mannheim, "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around that time, as engineering became a recognized professional activity, that slide rules came into wide use in Europe. They did not become common in the United States until 1881, when Edwin Thacher introduced a cylindrical rule there. The duplex rule was invented by William Cox in 1891, and was produced by Keuffel and Esser Co. of New York.^{[7]}^{[8]}
Astronomical work also required fine computations, and in 19th century Germany a steel slide rule about 2 meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.
Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession (in the same way that the stethoscope symbolizes the medical profession).^{[citation needed]} German rocket scientist Wernher von Braun brought two 1930s vintage Nestler slide rules with him when he moved to the U.S. after World War II to work on the American space program. Throughout his life he never used any other pocket calculating devices; slide rules served him perfectly well for making quick estimates of rocket design parameters and other figures. Aluminium Pickettbrand slide rules were carried on five Apollo space missions, including to the moon, according to advertising on Pickett's N600 slide rule boxes.^{[9]}
Some engineering students and engineers carried teninch slide rules in belt holsters, and even into the mid 1970s this was a common sight on campuses. Students also might keep a ten or twentyinch rule for precision work at home or the office ^{[10]} while carrying a fiveinch pocket slide rule around with them.
In 2004, education researchers David B. Sher and Dean C. Nataro conceived a new type of slide rule based on prosthaphaeresis, an algorithm for rapidly computing products that predates logarithms. There has been little practical interest in constructing one beyond the initial prototype, however. ^{[11]}
Slide rules have often been specialized to varying degrees for their field of use, such as excise, proof calculation, engineering, navigation, etc., but some slide rules are extremely specialized for very narrow applications. For example, the John Rabone & Sons 1892 catalog lists a "Measuring Tape and Cattle Gauge", a device to estimate the weight of a cow from its measurements.
There were many specialized slide rules for photographic applications; for example, the actinograph of Hurter and Driffield was a twoslide boxwood, brass, and cardboard device for estimating exposure from time of day, time of year, and latitude.
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.^{[12]}
In World War II, bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the U.S. Navy actually designed a generic slide rule "chassis" with an aluminium body and plastic cursor into which celluloid cards (printed on both sides) could be placed for special calculations. The process was invented to calculate range, fuel use and altitude for aircraft, and then adapted to many other purposes.
The importance of the slide rule began to diminish as electronic computers, a new but very scarce resource in the 1950s, became widely available to technical workers during the 1960s. The introduction of Fortran in 1957 made computers practical for solving modest size mathematical problems. IBM introduced a series of more affordable computers, the IBM 650 (1954), IBM 1620 (1959), IBM 1130 (1965) addressed to the science and engineering market. The BASIC programming language (1964) made it easy for students to use computers. The DEC PDP8 minicomputer was introduced in 1965.
Computers also changed the nature of calculation. With slide rules, there was a great emphasis on working the algebra to get expressions into the most computable form. Users of slide rules would simply approximate or drop small terms to simplify the calculation. Fortran allowed complicated formulas to be typed in from textbooks without the effort of reformulation. Numerical integration was often easier than trying to find closed form solutions for difficult problems. The young engineer asking for computer time to solve a problem that could have been done by a few swipes on the slide rule became a humorous cliché. Many computer centers had a framed slide rule hung on a wall with the note "In case of emergency, break glass."
Another step toward the replacement of slide rules with electronics was the development of electronic calculators for scientific and engineering use. The first included the Wang Laboratories LOCI2,^{[13]}^{[14]} introduced in 1965, which used logarithms for multiplication and division and the HewlettPackard HP9100, introduced in 1968.^{[15]} The HP9100 had trigonometric functions (sin, cos, tan) in addition to exponentials and logarithms. It used the CORDIC (coordinate rotation digital computer) algorithm,^{[16]} which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators.
The era of the slide rule ended with the launch of pocketsized scientific calculators, of which the 1972 HewlettPackard HP35 was the first. Such calculators became known as "slide rule" calculators, since they could perform most, or all the functions of a slide rule. Introduced at US$395, even this was considered expensive for most students. But by 1975, basic fourfunction electronic calculators could be purchased for less than $50. By 1976 the TI30 offered a scientific calculator for less than $25. After this time, the market for slide rules dwindled quickly as small scientific calculators became affordable.
An advantage of using a slide rule together with an electronic calculator is that an important calculation can be checked by doing it on both; because the two instruments are so different, there is little chance of making the same mistake twice.
For many of these reasons slide rules are still commonly used in aviation, particularly for smaller planes. They only being replaced by integrated, special purpose and expensive flight computers, and not general purpose calculators.
There are still people who prefer a slide rule over an electronic calculator as a practical computing device. Many others keep their old slide rules out of a sense of nostalgia, or collect slide rules as a hobby.^{[17]}
A popular collectible model is the Keuffel & Esser DeciLon, a premium scientific and engineering slide rule available both in a teninch "regular" (DeciLon 10) and a fiveinch "pocket" (DeciLon 5) variant. Another prized American model is the eightinch Scientific Instruments circular rule. Of European rules, FaberCastell's highend models are the most popular among collectors.
Although there is a large supply of slide rules circulating on the market, specimens in good condition tend to be surprisingly expensive. Many rules found for sale on online auction sites are damaged or have missing parts, and the seller may not know enough to supply the relevant information. Replacement parts are scarce, and therefore expensive, and are generally only available for separate purchase on individual collectors' web sites. The Keuffel and Esser rules from the period up to about 1950 are particularly problematic, because the endpieces on the cursors, made of celluloid, tend to break down chemically over time.
In many cases, an economical method for obtaining a working slide rule is to buy more than one of the same model, and combine their parts.
There are still a handful of sources for brand new slide rules. The Concise Company of Tokyo, which began as a manufacturer of circular slide rules in July 1954,^{[18]} continues to make and sell them today. And in September 2009, online retailer ThinkGeek introduced its own brand of straight slide rules, which they describe as "faithful replica[s]" that are "individually hand tooled" due to a stated lack of any existing manufacturers.^{[19]} The E6B circular slide rule used by pilots has been in continuous production and remains available in a variety of models. Proportion wheels are still used in graphic design.
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The slide rule, also known as a slipstick^{[1]}, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for "scientific" functions such as roots, logarithms and trigonometry, but does not generally perform addition or subtraction.
There are many different styles of slide rules. Generally, they are either have a linear form or that of a circle. They have a standardised set of markings (called scales). These scales are used for mathematical computations. Some slide rules have been made for specialised fields of application, for example aviation or finance. Such slide rules have special scales which are useful in the particular field of application, in addition to the common ones.
William Oughtred and others developed the slide rule in the 1600s. The slide rule is based on the work on logarithms by John Napier. Before electronic calculators were developed, slide rules were the tool used most often in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were gradually introduced; but around 1974 the electronic scientific calculator made the slide rule largely obsolete and most suppliers exited the business.
Contents 
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 timeconsuming and errorprone when done on paper. More complex slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.
Mathematical calculations are done by aligning a mark on the sliding central strip with one on one of the fixes strips. The relative position of other marks can then be observed. 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 have three linear strips of the same length. The strips are aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change.
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, 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 next 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 $\backslash log(xy)\; =\; \backslash log(x)\; +\; \backslash log(y)$ and $\backslash log(x/y)\; =\; \backslash log(x)\; \; \backslash log(y)$. Moving the top scale to the right by a distance of $\backslash log(x)$, by matching the beginning of the top scale with the label $x$ on the bottom, aligns each number $y$, at position $\backslash log(y)$ on the top scale, with the number at position $\backslash log(x)\; +\; \backslash log(y)$ on the bottom scale. Because $\backslash log(x)\; +\; \backslash log(y)\; =\; \backslash log(xy)$, this position on the bottom scale gives $xy$, the product of $x$ and $y$. 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 offscale 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 offscale, 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 (log10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (e^{x}) 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:
A, B  twodecade logarithmic scales, used for finding square roots and squares of numbers 
C, D  singledecade logarithmic scales 
K  threedecade logarithmic scale, used for finding cube roots and cubes of numbers 
CF, DF  "folded" versions of the C and D scales that start from π rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 but 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, DIF  "inverted" scales, running from right to left, used to simplify 1/x steps 
S  used for finding sines and cosines on the D scale 
T  used for finding tangents and cotangents on the D and DI scales 
ST, SRT  used for sines and tangents of small angles and degree–radian conversion 
L  a linear scale, used along with the C and D scales for finding base10 logarithms and powers of 10 
LLn  a set of loglog scales, used for finding logarithms and exponentials of numbers 
Ln  a linear scale, used along with the C and D scales for finding natural (base e) logarithms and $e^x$ 
 
The scales on the front and back of a K&E 40813 slide rule. 
The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions. ^{[2]}
There are singledecade (C and D), doubledecade (A and B), and tripledecade (K) scales. To compute $x^2$, 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 $x^y$ 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 $x^\{y/2\}$ and square it using the A and B scales as described above.
The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.
For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C. The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with C or, for angles greater than 45 degrees, CI. Common forms such as $k\backslash sin\; x$ can be read directly from x on the S scale to the result on the D scale, when the Cscale index is set at k. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/radian. Inverse trigonometric functions are found by reversing the process.
Many slide rules have S, T, and ST scales marked with degrees and minutes. Socalled decitrig models use decimal fractions of degrees instead.
Base10 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^{x} on the C (or D) scale and e^{–x} 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.^{[needs proof]}
Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques.^{[3]}
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 highspeed 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 "10inch" 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. [1]
Typically the divisions mark a scale to a precision of two significant figures, and the user estimates the third figure. Some highend slide rules have magnifying cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10inch slide rule to serve as well as a 20inch.
Various other conveniences have been developed. Trigonometric scales are sometimes duallabeled, in black and red, with complementary angles, the socalled "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.^{[needs proof]}
[[File:thumbA simple circular slide rule, made by Concise Co., Ltd., Tokyo, Japan, with only inverse, square and cubic scales. On the reverse is a handy list of 38 metric/imperial conversion factors.]] [[File:thumbleftBreitling Navitimer wristwatch with circular slide rule.]] 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, a 10 cm circular would have a maximum precision equal to a 30 cm ordinary slide rule. Circular slide rules also eliminate "offscale" calculations, because the scales were designed to "wrap around"; they never have to be reoriented when results are near 1.0—the rule is always on scale. However, for noncyclical nonspiral scales such as S, T, and LL's, the scale length is shortened to make room for end margins.^{[4]}
Circular slide rules are mechanically more rugged and smoothermoving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm offcentre of the pivot can result in a 0.2 mm worst case alignment error. The pivot, however, does prevent scratching of the face and cursors. The highest accuracy scales are placed on the outer rings. Rather than "split" scales, highend circular rules use spiral scales for more complex operations like logoflog scales. One eightinch premium circular rule had a 50inch spiral loglog scale.
The main disadvantages of circular slide rules are the difficulty in locating figures along a rotating disc, and limited number of scales. Another drawback of circular slide rules is that lessimportant scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch.
One slide rule remaining in daily use around the world is the E6B. This is a circular slide rule first created in the 1930s for aircraft pilots to help with dead reckoning. With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The socalled "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions, the E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of its mastery.
In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer–nautical mile and gallon–liter fuel amount conversion functions.
Traditionally slide rules were made out of hard wood such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument was made of steel.
In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable, strong and naturally selflubricating. These bamboo slide rules were introduced in Sweden in September, 1933 [2], and probably only a little earlier in Germany. Scales were made of celluloid or plastic. Later slide rules were made of plastic, or aluminum painted with plastic. Later cursors were acrylics or polycarbonates sliding on Teflon bearings.
All premium slide rules had numbers and scales engraved, and then filled with paint or other resin. Painted or imprinted slide rules were viewed as inferior because the markings could wear off. Nevertheless, Pickett, probably America's most successful slide rule company, made all printed scales. Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. The recommended cleaning method for engraved markings is to scrub lightly with steelwool. For painted slide rules, and the faint of heart, use diluted commercial windowcleaning fluid and a soft cloth.
[[File:thumbrightWilliam Oughtred (1575–1660), inventor of the circular slide rule.]] The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593  1656), an English Mathematician, in a book entitled “L'usage de la reigle de proportion en l'arithmetique & geometrie”. The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 he combined two Gunter rules, held together with the hands, to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students, but delayed in publishing them, and like Newton, he became involved in a vitriolic controversy over priority, with his onetime student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.
In 1677, Henry Coggeshall created a twofoot folding rule for timber measure, called the Coggeshall slide rule. His design and uses for the tool gave the slide rule purpose outside of mathematical inquiry.
In 1722, Warner introduced the two and threedecade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.
In 1815, Peter Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.
The more modern form was created in 1859 by French artillery lieutenant Amédée Mannheim, "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around that time, as engineering became a recognized professional activity, that slide rules came into wide use in Europe. They did not become common in the United States until 1881, when Edwin Thacher introduced a cylindrical rule there. The duplex rule was invented by William Cox in 1891, and was produced by Keuffel and Esser Co. of New York.^{[5]}^{[6]}
Astronomical work also required fine computations, and in 19th century Germany a steel slide rule about 2 meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.
In World War II, bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the U.S. Navy actually designed a generic slide rule "chassis" with an aluminum body and plastic cursor into which celluloid cards (printed on both sides) could be placed for special calculations. The process was invented to calculate range, fuel use and altitude for aircraft, and then adapted to many other purposes.
Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession (in the same way that the stethoscope symbolizes the medical profession).^{[needs proof]} German rocket scientist Wernher von Braun brought two 1930s vintage Nestler slide rules with him when he moved to the U.S. after World War II to work on the American space program. Throughout his life he never used any other pocket calculating devices; slide rules served him perfectly well for making quick estimates of rocket design parameters and other figures. Aluminum Pickettbrand slide rules were carried on five Apollo space missions, including to the moon, according to advertising on Pickett's N600 slide rule boxes [3].
Some engineering students and engineers carried teninch slide rules in belt holsters, and even into the mid 1970s this was a common sight on campuses. Students also might keep a tenor twentyinch rule for precision work at home or the office while carrying a fiveinch pocket slide rule around with them.
In 2004, education researchers David B. Sher and Dean C. Nataro conceived a new type of slide rule based on prosthaphaeresis, an algorithm for rapidly computing products that predates logarithms. There has been little practical interest in constructing one beyond the initial prototype, however. [4]
The importance of the slide rule began to diminish as electronic computers, a new but very scarce resource in the 1950s, became widely available to technical workers during the 1960s. The introduction of Fortran in 1957 made computers practical for solving modest size mathematical problems. IBM introduced a series of more affordable computers, the IBM 650 (1954), IBM 1620 (1959), IBM 1130 (1965) addressed to the science and engineering market. John Kemeny's BASIC programming language (1964) made it easy for students to use computers. The DEC PDP8 minicomputer was introduced in 1965.
Computers also changed the nature of calculation. With slide rules, there was a great emphasis on working the algebra to get expressions into the most computable form. Users of slide rules would simply approximate or drop small terms to simplify the calculation. Fortran allowed complicated formulas to be typed in from textbooks without the effort of reformulation. Numerical integration was often easier than trying to find closed form solutions for difficult problems. The young engineer asking for computer time to solve a problem that could have been done by a few swipes on the slide rule became a humorous cliché. Many computer centers had a framed slide rule hung on a wall with the note "In case of emergency, break glass."
Another step toward the replacement of slide rules with electronics was the development of electronic calculators for scientific and engineering use. The first included the Wang Laboratories LOCI2, ^{[7]} introduced in 1965, which used logarithms for multiplication and division and the HewlettPackard HP9100, introduced in 1968. ^{[8]} The HP9100 had trigonometric functions (sin, cos, tan) in addition to exponentials and logarithms. It used the CORDIC (coordinate rotation digital computer) algorithm, ^{[9]} which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators.
The last nail in the coffin for the slide rule was the launch of pocketsized scientific calculators, of which the 1972 HewlettPackard HP35 was the first. Such calculators became known as "slide rule" calculators since they could perform most or all of the functions on a slide rule. At several hundred dollars, even this was considered expensive for most students. While professional slide rules could also be quite expensive, drug stores often sold basic plastic models for under $20 USD. But by 1975, basic fourfunction electronic calculators could be purchased for under $50. By 1976 the TI30 offered a scientific calculator for under $25. After this time, the market for slide rules dried up quickly as small scientific calculators became affordable.
One advantage of using a slide rule together with an electronic calculator is that an important calculation can be checked by doing it on both; because the two instruments are so different, there is little chance of making the same mistake twice.
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