45-522: Otis Carter Formby King (1876–1944) was an electrical engineer in London who invented and produced a cylindrical slide rule with helical scales, primarily for business uses initially. The product was named Otis King's Patent Calculator, and was manufactured and sold by Carbic Ltd. in London from about 1922 to about 1972. With a log-scale decade length of 66 inches, the Otis King calculator should be about
90-427: A lookup table that maps from position on the ruler as each function's input. Calculations that can be reduced to simple addition or subtraction using those precomputed functions can be solved by aligning the two rulers and reading the approximate result. For example, a number to be multiplied on one logarithmic-scale ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying
135-409: A 10 cm (3.9 in) circular would have a maximum precision approximately equal to a 31.4 cm (12.4 in) 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,
180-438: 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
225-443: A free dish and one cursor. The dual cursor versions perform multiplication and division by holding a constant 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,
270-518: A full digit more accurate than a 6-inch pocket slide rule. But due to inaccuracies in tic-mark placement, some portions of its scales will read off by more than they should. For example, a reading of 4.630 might represent an answer of 4.632, or almost one part in 2000 error, when it should be accurate to one part in 6000 (66"/6000 = 0.011" estimated interpolation accuracy). The Geniac brand cylindrical slide rule sold by Oliver Garfield Company in New York
315-399: 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. Scales may be grouped in decades , where each decade corresponds to a range of numbers that spans a ratio of 10 (i.e. a range from 10 to 10 ). For example,
360-564: Is a hand -operated mechanical calculator consisting of slidable rulers for evaluating mathematical operations such as multiplication , division , exponents , roots , logarithms , and trigonometry . It is one of the simplest analog computers . Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule
405-425: Is again positioned to start at the 2 on the bottom scale. Since 2 represents 20 , all numbers in that scale are multiplied by 10 . Thus, any answer in the second set of numbers is multiplied by 100 . Since 8.8 in the top scale represents 88 , the answer must additionally be multiplied by 10 . The answer directly reads 1.76 . Multiply by 100 and then by 10 to get the actual answer: 1,760 . In general,
450-464: Is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler , the slide rule is not meant to be used for measuring length or drawing straight lines. Nor is it designed for addition or subtraction, which is usually performed using other methods, like using an abacus . Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation
495-500: 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 aren't typically performed on slide rules, but
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#1732868939789540-613: Is possible using either of the following two techniques: Using (almost) any strictly monotonic scales , other calculations can also be made with one movement. For example, reciprocal scales can be used for the equality 1 x + 1 y = 1 z {\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}} (calculating parallel resistances , harmonic mean , etc.), and quadratic scales can be used to solve x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} . The width of
585-562: Is used to keep track of the order of magnitude of results. English mathematician and clergyman Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier . It made calculations faster and less error-prone than evaluating on paper . Before the advent of the scientific pocket calculator , it was the most commonly used calculation tool in science and engineering . The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through
630-452: 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 mark are proportional to the logarithms of the marked values. 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 resulting quotient, 2.75 , can then be read below
675-481: The law of the logarithm of a product , the product of the two numbers can be read. More elaborate slide rules can perform other calculations, such as square roots , exponentials , logarithms , and trigonometric functions . The user may estimate the location of the decimal point in the result by mentally interpolating between labeled graduations. Scientific notation is used to track the decimal point for more precise calculations. Addition and subtraction steps in
720-478: 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,
765-538: The 1950s and 1960s, even as desktop electronic computers were gradually introduced. But after the handheld scientific calculator was introduced in 1972 and became inexpensive in the mid-1970s, slide rules became largely obsolete , so most suppliers departed the business. In the United States , the slide rule is colloquially called a slipstick . Each ruler's scale has graduations labeled with precomputed outputs of various mathematical functions , acting as
810-406: The 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where 7 is on the top scale: [REDACTED] But since the 7 is above the second set of numbers that number must be multiplied by 10 . Thus, even though the answer directly reads 1.4 , the correct answer is 1.4×10 = 14 . For an example with even larger numbers, to multiply 88×20 , the top scale
855-419: 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 a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} can be solved by first reducing
900-579: The CI scale is used. Common forms such as k sin x {\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
945-466: 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 {\displaystyle x^{y}} problems, use
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#1732868939789990-411: 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 p {\displaystyle p} . These two values are the roots of the equation. The LLN scales can be used to compute and compare the cost or return on a fixed rate loan or investment. The simplest case is for continuously compounded interest. Example: Taking D as
1035-478: 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. Though not as common since
1080-492: 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 {\displaystyle x^{y/2}} and square it using the A and B scales as described above. Alternatively, use
1125-480: The S scale with C (or D) scale. (On many closed-body rules the S scale relates to the A and B scales instead and covers angles from around 0.57 up to 90 degrees; 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
1170-452: The act of positioning the top scale to start at the bottom scale's label for x {\displaystyle x} corresponds to shifting the top logarithmic scale by a distance of log ( x ) {\displaystyle \log(x)} . This aligns each top scale's number y {\displaystyle y} at offset log ( y ) {\displaystyle \log(y)} with
1215-661: The advent of computerized layout, they 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. Logarithmic-scale Too Many Requests If you report this error to
1260-564: The bottom scale's number at position log ( x ) + log ( y ) {\displaystyle \log(x)+\log(y)} . Because log ( x ) + log ( y ) = log ( x × y ) {\displaystyle \log(x)+\log(y)=\log(x\times y)} , the mark on the bottom scale at that position corresponds to x × y {\displaystyle x\times y} . With x=2 and y=3 for example, by positioning
1305-464: 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
1350-474: 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,
1395-423: The equation to the form x 2 − p x + q = 0 {\displaystyle x^{2}-px+q=0} (where p = − b / a {\displaystyle p=-b/a} and q = c / a {\displaystyle q=c/a} ), and then aligning the index ("1") of the C scale to the value q {\displaystyle q} on
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1440-520: The interest rate in percent, slide the index (the "1" at the right or left end of the scale) of C to the percent on D. The corresponding value on LL2 directly below the index will be the multiplier for 10 cycles of interest (typically years). The value on LL2 below 2 on the C scale will be the multiplier after 20 cycles, and so on. 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
1485-647: The log of a value on a multiplier scale), natural logarithm (ln) and exponential ( e ) scales. Others feature scales for calculating hyperbolic functions . On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order. The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions. There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute x 2 {\displaystyle x^{2}} , for example, locate x on
1530-593: The other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%. In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular are trigonometric , usually sine and tangent , common logarithm (log 10 ) (for taking
1575-458: 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
1620-404: 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
1665-888: The range 1 to 10 is a single decade, and the range from 10 to 100 is another decade. Thus, single-decade scales (named C and D) range from 1 to 10 across the entire length of the slide rule, while double-decade scales (named A and B) range from 1 to 100 over the length of the slide rule. The following logarithmic identities transform the operations of multiplication and division to addition and subtraction, respectively: log ( x × y ) = log ( x ) + log ( y ) , {\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} log ( x / y ) = log ( x ) − log ( y ) . {\displaystyle \log(x/y)=\log(x)-\log(y)\,.} With two logarithmic scales,
1710-456: The result, 8.25 , can be read beneath the 3 in the top scale in the figure above, without the need to register the intermediate result for 5.5 / 2 . Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems. For example, setting 7.5 on one scale over 10 on
1755-466: 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
1800-453: The scale width is narrowed to make room for end margins. 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 (0.0039 in) off-centre of the pivot can result in a 0.2 mm (0.0079 in) worst case alignment error. The pivot does prevent scratching of the face and cursors. The highest accuracy scales are placed on
1845-440: The slide rule is quoted in terms of the nominal width of the scales. Scales on the most common "10-inch" models are actually 25 cm, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflows. Pocket rules are typically 5 inches (12 cm). Models a couple of metres (yards) wide were made to be hung in classrooms for teaching purposes. Typically
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1890-582: 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. For example, instead of reading from an A scale to a D scale to find a square root, it may be possible to read from a D scale to an R1 scale running from 1 to square root of 10 or to an R2 scale running from square root of 10 to 10, where having more subdivisions marked can result in being able to read an answer with one more significant digit. Circular slide rules come in two basic types, one with two cursors, and another with
1935-405: The top scale to start at the bottom scale's 2 , the result of the multiplication 3×2=6 can then be read on the bottom scale under the top scale's 3 : [REDACTED] While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to 7×2=14 is found by first positioning the top scale to start above
1980-534: The top scale's 1 : [REDACTED] There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end. With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus 5.5×3 / 2 would be computed as 5.5 / 2 ×3 and
2025-443: Was initially a relabelled Otis King; Garfield later made his own, probably unauthorized version of the Otis King (around 1959). The UK patents covering the mechanical device(s) would have expired in about 1941–1942 (i.e. 20 years after filing of the patent) but copyright in the drawings- which would arguably include the spiral scale layout- would typically only expire 70 years after the author's death. Slide rule A slide rule
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