The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions.^[13]

Roots and powers

There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute ${\displaystyle x^{2}}$, 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 ${\displaystyle x^{y}}$ 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 ${\displaystyle x^{y/2}}$ 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 cu

There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute ${\displaystyle x^{2}}$, 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 ${\displaystyle {}^{}}$

For ${\displaystyle x^{y}}$ 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 ${\displaystyle x^{y/2}}$ 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.)

Quadratic equations of the form ${\displaystyle ax^{2}+bx+c=0}$ can be solved by first reducing the equation to the form ${\displaystyle x^{2}-px+q=0}$ (where ${\displaystyle p=-b/a}$ and ${\displaystyle q=c/a}$), and then sliding the index of the C scale to the value ${\displaystyle q}$ on the D scale. The cursor is then moved along the rule until a position is found where the numbers on the CI and D scales add up to ${\displaystyle p}$. These two values are the roots of the equation.

Trigonometry

The S, T, and ST scales are used for trig fun

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 (or D) scale; though on many closed-body rules the S scale relates to the A scale instead, and what follows must be adjusted appropriately. The S scale has a second set of angles (sometimes in a different color), whic

For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C (or D) scale; though on many closed-body rules the S scale relates to the A scale instead, and what follows must be adjusted appropriately. 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 the C (or D) scale for angles less than 45 degrees. For angles greater than 45 degrees the CI scale is used. Common forms such as ${\displaystyle k\sin x}$ can be read directly from x on the S scale to the result on the D scale, when the C-scale 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 (e.g. some Keuffel and Esser models (Doric duplex 5" models, for example), late-model Teledyne-Post Mannheim-type rules). 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. Logarithms to any other base can be calculated by reversing the procedure for calculating powers of a number. For example, log2 values can be determined by lining up either leftmost or rightmost 1 on the C scale with 2 on the LL2 scale, finding the number whose logarithm is to be calculated on the corresponding LL scale, and reading the log2 value on the C scale.

Addition and subtraction

Slide

Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques.^[14]

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:

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:

${\displaystyle x-y=\left({\frac {x}{y}}-1\right)y.}$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.

Generalizations

<

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.

Using (almost) any strictly monotonic scales, other calculations can also be made with one movement.^[15][16] For example, reciprocal scales can be used for the equality ${\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}}$(calculating parallel resistances, harmonic mean, etc.), and quadratic scales can be used to solve ${\displaystyle x^{2}+y^{2}=z^{2}}$.

Physical design

Standard linear rules

[17]

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 magnifier 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 model.

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.^{[further explanation needed]}

Circular slide rules

Circular slide rules come in two basic types, one with two cursors, and another with a free dish and one cursor. The dual cursor versions perform multiplication and division by holding a fast angle between the cursors as they are rotated around

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 magnifier 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 model.

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.^{[further explanation needed]}

Circular slide rules come in two basic types, one with two cursors, and another with a free dish and one cursor. The dual cursor versions perform multiplication and division by holding a fast angle between the cursors as they are rotated around the dial. The onefold 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 widest 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 approximately equal to a 31.4 cm ordinary slide rule. Circular slide rules also eliminate "off-

The basic advantage of a circular slide rule is that the widest 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 approximately equal to a 31.4 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 reoriented 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 width is narrowed to make room for end margins.^[18]

Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm off-centre 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, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale. Around 1970, an inexpensive model from B. C. Boykin (Model 510) featured 20 scales, including 50-inch C-D (multiplication) and log scales. The RotaRule featured a friction brake for the cursor.

The main disadvantages of circular slide rules are the difficulty in locating figures along a dish, and limited number of scales. Another drawback of circular slide rules is that less-important 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 so-called "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 calculate aspect ratios. Lining up the original and desired size values on the inner and outer wheels will display their ratio as a percentage in a small window. They are not as common since the advent of computerized layout, but are still made and used.^{[citation needed]}

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.

A 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.

A two-scale slide rule built into

A two-scale slide rule built into a ring

Breitling Navitimer wristwatch with circular slide rule

The front side of a Boykin R

The front side of a Boykin RotaRule Model 510

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The rear side of a Boykin RotaRule Model 510

Cylindrical slide rules

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 potent

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 greater precision, than afforded by a straight or circular rule.

• 

  Materials

  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 self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933,[19] and probably only a little earlier in Germany. Scales

  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 self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933,^[19] and p

  In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable, strong, and naturally self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933,^[19] and probably only a little earlier in Germany. Scales were made of celluloid, plastic, or painted aluminium. 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^{[citation needed]} 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 slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. In 1620 Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide.^[20] In c. 1622, William Oughtred of Cambridge combined two handheld Gunter rules to make a device that is recognizably the modern slide rule.^[21] Oughtred became involved in a vitriolic controversy over priority, with his one-time 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 two-foot folding rule for timber measure, called the Coggeshall slide rule, expanding the slide rule's use beyond mathematical inquiry.

  In 1722, Warner introduced the two- and three-decade 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.

  In 1821, Nathaniel Bowditch, described in the American Practical Navigator a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.

  In 1845, Paul Cameron of Glasgow introduced a nautical slide rule capable of answering navigation questions, including right ascension and declination of the sun and principal stars.^[22]

  Modern form

  The importance of the slide rule began to diminish as electronic computers, a new but rare resource in the 1950s, became more widely available to technical workers during the 1960s. (See History of computing hardware (1960s–present).)

  Another step away from slide rules was the introduction of relatively inexpensive electronic desktop scientific calculators. The first included the Wang Laboratories LOCI-2,^[34][35] introduced in 1965, which used logarithms for multiplication and division; and the Hewlett-Packard HP 9100A, introduced in 1968.^[36] Both of these were programmable and provided exponential and logarithmic functions; the HP had Wang Laboratories LOCI-2,^[34][35] introduced in 1965, which used logarithms for multiplication and division; and the Hewlett-Packard HP 9100A, introduced in 1968.^[36] Both of these were programmable and provided exponential and logarithmic functions; the HP had trigonometric functions (sine, cosine, and tangent) and hyperbolic trigonometric functions as well. The HP used the CORDIC (coordinate rotation digital computer) algorithm,^[37] which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators.

  As with mainframe computing, the availability of these machines did not significantly affect the ubiquitous use of the slide rule until cheap hand held scientific electronic calculators became available in the mid-1970s, at which point, it rapidly declined. The pocket-sized Hewlett-Packard HP-35 scientific calculator was the first handheld device of its type, but it cost US$395 in 1972. This was justifiable for some engineering professionals but too expensive for most students. By 1975, basic four-function electronic calculators could be purchased for less than $50, and by 1976 the TI-30 scientific calculator was sold for less than $25 ($112 adjusted for inflation).

  Most people^{[citation needed]} find slide rules difficult to understand and use. Even during their heyday, they never caught on with the general public.^[38] Addition and subtraction are not well-supported operations on slide rules and doing a calculation on a slide rule tends to be slower than on a calculator.^[39] This led engineers to use mathematical equations that favored operations that were easy on a slide rule over more accurate but complex functions; these approximations could lead to inaccuracies and mistakes.^[40] On the other hand, the spatial, manual operation of slide rules cultivates in the user an intuition for numerical relationships and scale that people who have used only digital calculators often lack.^[41] A slide rule will also display all the terms of a calculation along with the result, thus eliminating uncertainty about what calculation was actually performed.

  A slide rule requires the user to separately compute the order of magnitude of the answer in order to position the decimal point in the results. For example, 1.5 × 30 (which equals 45) will show the same result as 1,500,000 × 0.03 (which equals 45,000). This separate calculation is less likely to lead to extreme calculation errors, but forces the user to keep track of magnitude in short-term memory (which is error-prone), keep notes (which is cumbersome) or reason about it in every step (which distracts from the other calculation requirements).

  The typical arithmetic precision of a slide rule is about three significant digits, compared to many digits on digital calculators. As order of magnitude gets the greatest prominence when using a slide rule, users are less likely to make errors of false precision.

  When performing a sequence of multiplications or divisions by the same number, the answer can often be determined by merely glancing at the slide rule without any manipulation. This can be especially useful when calculating percentages (e.g. for test scores) or when comparing prices (e.g. in dollar

  A slide rule requires the user to separately compute the order of magnitude of the answer in order to position the decimal point in the results. For example, 1.5 × 30 (which equals 45) will show the same result as 1,500,000 × 0.03 (which equals 45,000). This separate calculation is less likely to lead to extreme calculation errors, but forces the user to keep track of magnitude in short-term memory (which is error-prone), keep notes (which is cumbersome) or reason about it in every step (which distracts from the other calculation requirements).

  The typical arithmetic precision of a slide rule is about three significant digits, compared to many digits on digital calculators. As order of magnitude gets the greatest prominence when using a slide rule, users are less likely to make errors of false precision.

  When performing a sequence of multiplications or divisions by the same number, the answer can often be determined by merely glancing at the slide rule without any manipulation. This can be especially useful when calculating percentages (e.g. for test scores) or when comparing prices (e.g. in dollars per kilogram). Multiple speed-time-distance calculations can be performed hands-free at a glance with a slide rule. Other useful linear conversions such as pounds to kilograms can be easily marked on the rule and used directly in calculations.

  Being entirely mechanical, a slide rule does not depend on grid electricity or batteries. However, mechanical imprecision in slide rules that were poorly constructed or warped by heat or use will lead to errors.

  Many sailors keep slide rules as backups for navigation in case of electric failure or battery depletion on long route segments. Slide rules are still commonly used in aviation, particularly for smaller planes. They are being replaced only by integrated, special purpose and expensive flight computers, and not general-purpose calculators. The E6B circular slide rule used by pilots has been in continuous production and remains available in a variety of models. Some wrist watches designed for aviation use still feature slide rule scales to permit quick calculations. The Citizen Skyhawk AT and the Seiko Flightmaster SNA411 are two notable examples.^[42]

  Even in the 2000s, some people preferred a slide rule over an electronic calculator as a practical computing device. Others kept their old slide rules out of a sense of nostalgia, or collected them as a hobby.^[43]

  A popular collectible model is the Keuffel & Esser Deci-Lon, a premium scientific and engineering slide rule available both in a ten-inch (25 cm) "regular" (Deci-Lon 10) and a five-inch "pocket" (Deci-Lon 5) variant. Another prized American model is the eight-inch (20 cm) Scientific Instruments circular rule. Of European rules, Faber-Castell's high-end models are the most popular among collectors.

  Although a great many slide rules are circulating on the market, specimens in good condition tend to be 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, expensive, and generally available only 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 end-pieces on the cursors, made of celluloid, tend to chemically break down over time.

  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,^[44] continues to make and sell them today. In September 2009, on-line retailer ThinkGeek introduced its own brand of straight slide rules, described as "faithful replica[s]" that are "individually hand tooled".^[45] These are no longer available in 2012.^[46] In addition, Faber-Castell had a number of slide rules in inventory, available for international purchase through their web store, until mid 2018.^[47] Proportion wheels are still used in graphic design.

  Various slide rule simulator apps are available for Android and iOS-based smart phones and tablets.

  Specialized slide rules such as the E6B used in aviation, and gunnery slide rules used in laying artillery are still used though no longer on a routine basis. These rules are used as part of the teaching and instruction process as in learning to use them the student also learns about the principles behind the calculations, it also allows the student to

  A popular collectible model is the Keuffel & Esser Deci-Lon, a premium scientific and engineering slide rule available both in a ten-inch (25 cm) "regular" (Deci-Lon 10) and a five-inch "pocket" (Deci-Lon 5) variant. Another prized American model is the eight-inch (20 cm) Scientific Instruments circular rule. Of European rules, Faber-Castell's high-end models are the most popular among collectors.

  Although a great many slide rules are circulating on the market, specimens in good condition tend to be 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, expensive, and generally available only 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 end-pieces on the cursors, made of celluloid, tend to chemically break down over time.

  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,^[44] continues to make and sell them today. In September 2009, on-line retailer ThinkGeek introduced its own brand of straight slide rules, described as "faithful replica[s]" that are "individually hand tooled".^[45] These are no longer available in 2012.^[46] In addition, Faber-Castell had a number of slide rules in inventory, available for international purchase through their web store, until mid 2018.^[47] Proportion wheels are still used in graphic design.

  Various slide rule simulator apps are available for Android and iOS-based smart phones and tablets.

  Specialized slide rules such as the E6B used in aviation, and gunnery slide rules used in laying artillery are still used though no longer on a routine basis. These rules are used as part of the teaching and instruction process as in learning to use them the student also learns about the principles behind the calculations, it also allows the student to be able to use these instruments as a back up in the event that the modern electronics in general use fail.

  The MIT Museum in Cambridge, Massachusetts, has a collection of hundreds of slide rules, nomograms, and mechanical calculators. The Keuffel and Esser Company slide rule collection, from the slide rule manufacturer formerly located in Brooklyn, New York, was donated to MIT around 2005.^[48] Selected items from the collection are usually on display at the Museum.^[49][50]

  See also