Slide rule - Knowledge

2429: 2526: 2995: 4586:. "With computation moving literally at a hand's pace and the lack of precision a given, mathematicians worked to simplify complex problems. Because linear equations were friendlier to slide rules than more complex functions were, scientists struggled to linearize mathematical relations, often sweeping high-order or less significant terms under the computational carpet. So a car designer might calculate gas consumption by looking mainly at an engine's power, while ignoring how air friction varies with speed. Engineers developed shortcuts and rules of thumb. At their best, these measures led to time savings, insight and understanding. On the downside, these approximations could hide mistakes and lead to gross errors." 2598: 2983: 2449: 4603:. "One effect was that users felt close to the numbers, aware of rounding-off errors and systematic inaccuracies, unlike users of today's computer-design programs. Chat with an engineer from the 1950s, and you will most likely hear a lament for the days when calculation went hand-in-hand with deeper comprehension. Instead of plugging numbers into a computer program, an engineer would understand the fine points of loads and stresses, voltages and currents, angles and distances. Numeric answers, crafted by hand, meant problem solving through knowledge and analysis rather than sheer number crunching." 2473: 2485: 3318: 4569:, Barbara Duch, Susan Groh, Deborah Allen, eds., Stylus Publishing, LLC, 2001. "Numerical computations in freshman physics and chemistry were excruciating; however, this did not seem to be the case for those students fortunate enough to already own a calculator. I vividly recall that at the end of 1974, the students who were still using slide rules were given an additional 15 minutes on the final examination to compensate for the computational advantage afforded by the calculator, hardly adequate compensation in the opinions of the remaining slide rule practitioners." 2574: 2953: 3098:

operations that were easy on a slide rule over more accurate but complex functions; these approximations could lead to inaccuracies and mistakes. 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. 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. It has thus been compared with

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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.

40: 2562: 2461: 2365:). For example, 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, the scale width is narrowed to make room for end margins. 2765:, who was fortunate both in having his rule made by a firm of national reputation, and its adoption by the French Artillery. Mannheim's rule had two major modifications that made it easier to use than previous general-purpose slide rules. Such rules had four basic scales, A, B, C, and D, and D was the only single-decade logarithmic scale; C had two decades, like A and B. Most operations were done on the A and B scales; D was only used for finding squares and square roots. 1546: 2668: 1553: 2965: 936: 4553:. Yes, the occasional grocery store manager figured discounts on a slipstick, and this author once caught his high school English teacher calculating stats for trifecta horse-race winners on a slide rule during study hall. But slide rules never made it into daily life because you could not do simple addition and subtraction with them, not to mention the difficulty of keeping track of the decimal point. Slide rules remained tools for techies." 2586: 2185: 268: 3179: 2941: 1140:) rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 and 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. 310:, 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, 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. 1609:, 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. 3017: 2648:. Painted or imprinted slide rules were viewed as inferior, because the markings could wear off or be chemically damaged. Nevertheless, Pickett, an American slide rule company, made only printed scale rules. Premium slide rules included clever mechanical catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. 2657: 181: 2769:

square roots as part of a larger calculation. Mannheim's rule also had a cursor, unlike almost all preceding rules, so any of the scales could be easily and accurately compared across the rule width. The "Mannheim rule" became the standard slide rule arrangement for the later 19th century and remained a common standard throughout the slide-rule era.

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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.

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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

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The width of 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

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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 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

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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

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introduced its own brand of straight slide rules, described as "faithful replica" that were "individually hand tooled". These were no longer available in 2012. In addition, Faber-Castell had a number of slide rules in inventory, available for international purchase through their web store, until mid

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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

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In 1881, the American inventor Edwin Thacher introduced his cylindrical rule, which had a much longer scale than standard linear rules and thus could calculate to higher precision, about four to five significant digits. However, the Thacher rule was quite expensive, as well as being non-portable, so

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For angles from around 5.7 up to 90 degrees, sines are found by comparing 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

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For example, setting 7.5 on one scale over 10 on 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

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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 backup in the

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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

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Some engineering students and engineers carried ten-inch slide rules in belt holsters, a common sight on campuses even into the mid-1970s. Until the advent of the pocket digital calculator, students also might keep a ten- or twenty-inch rule for precision work at home or the office while carrying a

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for two or three months. The simultaneous equation contained up to seven unknown quantities, took about a week to solve, and had to be repeated with a different selection of slack wires if the guess on which of the eight radial wires were slack was wrong and one of the wires guessed to be slack was

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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 constant angle between the cursors as they are rotated around the dial. The onefold cursor version operates more

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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

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As with mainframe computing, the availability of these desktop 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

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Mannheim changed the C scale to a single-decade scale and performed most operations with C and D instead of A and B. Because the C and D scales were single-decade, they could be read more precisely, so the rule's results could be more accurate. The change also made it easier to include squares and

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Slide rules have often been specialized to varying degrees for their field of use, such as excise, proof calculation, engineering, navigation, etc., and some slide rules are extremely specialized for very narrow applications. For example, the John Rabone & Sons 1892 catalog lists a "Measuring

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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

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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

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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

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At least one circular rule, a 1931 Gilson model, sacrificed some of the scales usually found in slide rules in order to obtain additional resolution in multiplication and division. It functioned through the use of a spiral C scale, which was claimed to be 50 feet and readable to five significant

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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

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Even during their heyday, slide rules never caught on with the general public. 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. This led engineers to use mathematical equations that favored

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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

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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. For

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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

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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.

2384:. 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 1690:

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.

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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

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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.

3126:). This separate calculation 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). 3292:

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, continues to make and sell them today. In September 2009, on-line retailer

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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.

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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.

2626:-clad 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, and probably only a little earlier in Germany. 1019:

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.

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Astronomical work also required precise computations, and, in 19th-century Germany, a steel slide rule about two meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.

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is claimed to be " most extensive resource for all things concerning slide rules and logarithmic calculators". The museum's Web page includes extensive literature relative to slide rules in its "Slide Rule Library" section.

3050:(coordinate rotation digital computer) algorithm, which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators. 2776:

profession during the later 19th century drove widespread slide-rule use, beginning in Europe and eventually taking hold in the United States as well. The duplex rule was invented by William Cox in 1891 and was produced by

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Around 1974, lower-cost handheld electronic scientific calculators started to make slide rules largely obsolete. By 1975, basic four-function electronic calculators could be purchased for less than $ 50, and by 1976 the

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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.

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This slide rule is positioned to yield several values: From C scale to D scale (multiply by 2), from D scale to C scale (divide by 2), A and B scales (multiply and divide by 4), A and D scales (squares and square

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Even in the 21st century, some people prefer a slide rule over an electronic calculator as a practical computing device. Others keep their old slide rules out of a sense of nostalgia, or collect them as a hobby.

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A Department of Defense publication from 1962 infamously included a special-purpose circular slide rule for calculating blast effects, overpressure, and radiation exposure from a given yield of an atomic bomb.

2555:, 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. 2396:. 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 the advent of computerized layout, they 1981:

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

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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.

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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.

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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

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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.

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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.

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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.

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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

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is used to track the decimal point for more precise calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.

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Pilot balloon slide rules were used by meteorologists in weather services to determine the upper wind velocities from an ascending hydrogen or helium-filled pilot balloon.

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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.

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scales, single sections of the same length, used together for multiplication and division, and generally one of them is combined with another scale for other calculations

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to use slide rules. The UIL had been originally been organized in 1910 to administer literary events, but had become the governing body of school sports events as well.

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marked with powers of 2, calculating 8x4 by aligning the bottom ruler to start where the top ruler is 8, and then reading at the number above where the bottom ruler is 4.

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a "sliding rule" that contained scaled trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.

3137:, 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 1964: 1688: 595: 543: 1840: 1393: 885:

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

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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

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of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide. In c. 1622,

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is found by first positioning the top scale to start above the 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where

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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

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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.)

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company distributed a customized slide rule adapted to solving the equations used for selecting the proper size of industrial flow control valves.

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The first step away from slide rules was the introduction of relatively inexpensive electronic desktop scientific calculators. These included the

4851: 3282: 2484: 145:. The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through the 1950s and 1960s, even as desktop 4622: 4084: 3272:

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

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Then in the early 1970s the first electronic hand-held calculators appeared on the market, and within ten years the slide rule was obsolete.

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are typically 5 inches (12 cm). Models a couple of metres (yards) wide were made to be hung in classrooms for teaching purposes.

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and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.

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A Russian circular slide rule built like a pocket watch that works as single cursor slide rule since the two needles are ganged together

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of the answer to position the decimal point in the results. For example, 1.5 × 30 (which equals 45) will show the same result as

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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

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scale, three sections each of which is one third the length of the C and D scales, used for finding cube roots and cubes of numbers

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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

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Addition and subtraction aren't typically performed on slide rules, but is possible using either of the following two techniques:

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scales, two sections each of which is half the length of the C and D scales, used for finding square roots and squares of numbers

409: 3265:) variant. Another prized American model is the eight-inch (20 cm) Scientific Instruments circular rule. Of European rules, 2952: 4060:"Lot 25368 Buzz Aldrin's Apollo 11 Slide Rule – Flown to the Moon. ... 2007 September Grand Format Air & Space Auction #669" 2501: 3156:

or batteries. Mechanical imprecision in slide rules that were poorly constructed or warped by heat or use will lead to errors.

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sheets with calculations "the truth stood revealed (and) produced a satisfaction almost amounting to a religious experience".

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After sliding the cursor right (for addition) or left (for subtraction) and returning the slide to 0, the result can be read.

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Pickett circular slide rule with two cursors. (4.25 in/10.9 cm width) Reverse has additional scale and one cursor.

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scales, for working with logarithms of any base and arbitrary exponents. 4, 6, or 8 scales of this type are commonly seen.

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

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In 1845, Paul Cameron of Glasgow introduced a nautical slide rule capable of answering navigation questions, including

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to make a device that is recognizably the modern slide rule. Oughtred became involved in a vitriolic controversy over

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used for addition, subtraction, and (along with the C and D scales) for finding base-10 logarithms and powers of 10

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Slide – a center strip interlocked with the frame that can move lengthwise relative to the frame.

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It is believed they may still have some slide rules, but their new website and online store does not carry them.

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Then, just a decade ago, the invention of the pocket calculator made the slide rule obsolete almost overnight...

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Wang Laboratories (December 1966). "Now you can determine Copolymer Composition in a few minutes at your desk".

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introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the

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Throughout the 1950s and 1960s, the slide rule was the symbol of the engineer's profession in the same way the

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Scales were also made of celluloid or other polymers, or printed on aluminium. Later cursors were molded from

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In 2004, education researchers David B. Sher and Dean C. Nataro conceived a new type of slide rule based on

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The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.

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With two logarithmic scales, the act of positioning the top scale to start at the bottom scale's label for

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that the quotient of two variables plus (or minus) one times the divisor equals their sum (or difference):

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interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%.

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Runner or glass – an exterior sliding piece with a hairline, also known as the "cursor".

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ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying the

4124:"The Prosthaphaeretic Slide Rule: A Mechanical Multiplication Device Based On Trigonometric Identities" 3257:, a premium scientific and engineering slide rule available both in a ten-inch (25 cm) "regular" ( 2260: 1493: 17: 4859: 5069: 4614: 4088: 2385: 2154: 2153:

This is similar to the addition/subtraction technique used for high-speed electronic circuits with a

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There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute

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

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All premium slide rules had numbers and scales deeply engraved, and then filled with paint or other

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and others developed the slide rule in the 17th century based on the emerging work on logarithms by

110:. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while 4063: 3333: 1709: 327:

transform the operations of multiplication and division to addition and subtraction, respectively:

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of the two numbers can be read. More elaborate slide rules can perform other calculations, such as

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Converting addition and subtraction to division (required for the C and D or comparable scales):

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from 1924. The stress calculations for each transverse frame required computations by a pair of

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used for application-specific computations. Though similar in name and appearance to a standard

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in the top scale in the figure above, without the need to register the intermediate result for

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Frame or base – two strips of the same length held parallel with a gap between.

245: 205: 71: 3831: 3687: 3498: 2940: 3429: 3375: 3341: 2778: 2707: 2410:. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured 1940: 1658: 568: 516: 229: 213: 150: 138: 56: 3046:(sine, cosine, and tangent) and hyperbolic trigonometric functions as well. The HP used the 2380:. This is a circular slide rule first created in the 1930s for aircraft pilots to help with 1808: 1368: 4009: 3866: 3709: 3130: 3035:

LOCI-2, introduced in 1965, which used logarithms for multiplication and division; and the

2970: 2898: 2762: 2585: 2543:

There are two main types of cylindrical slide rules: those with helical scales such as the

2162: 1845: 1615: 1585: 1456: 1398: 1084: 1076: 237: 5015: 3668: 3533: 3304:

Specialized slide rules such as the E6B used in aviation, and gunnery slide rules used in

1033: 8: 5084: 3774: 3408: 3349: 3317: 2706:

In 1677, Henry Coggeshall created a two-foot folding rule for timber measure, called the

2696: 2491: 2407: 2342: 1703: 1528: 256: 111: 4013: 3870: 3789: 3713: 3378: – Person performing mathematical calculations, before calculators became available 2893:

There were many specialized slide rules for photographic applications. For example, the

2890:

Tape and Cattle Gauge", a device to estimate the weight of a cow from its measurements.

4391: 4235: 4033: 3728: 3699: 3251: 3153: 3146: 3134: 3106: 2757:

Engineer using a slide rule, with mechanical calculator in background, mid 20th century

2725: 2552: 2193: 1899: 1879: 1436: 1267: 1211: 548: 496: 307: 197: 115: 4530:

The first hand calculator appeared in 1972 and made the slide rule obsolete overnight.

4021: 2392:

Proportion wheels are simple circular slide rules used in graphic design to calculate

180: 4519: 4509: 4478: 4450: 4383: 4227: 4041: 4025: 3978: 3874: 3837: 3588: 3563: 3504: 3468: 3345: 3032: 2831: 2692: 2544: 2319: 1643:

on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find

1565: 1072: 221: 4991: 4812: 4239: 3196: 2761:

A more modern form of slide rule was created in 1859 by French artillery lieutenant

4501: 4375: 4320: 4293: 4219: 4017: 3970: 3414: 3093:

explicitly comparing electronic computers to engineers calculating with slide rules

2877: 2717: 2700: 2688: 2660: 1970:

on the S scale to the result on the D scale, when the C scale index is set at

1261: 1064: 303: 193: 122: 60: 4961: 4644:"Citizen Watch Company – Citizen Eco-Drive / US, Canada, UK, IrelandCitizen Watch" 4395: 4339: 3647: 3235: 3066:

scientific calculator was sold for less than $ 25 ($ 134 adjusted for inflation).

1639:

problems, use the LL scales. When several LL scales are present, use the one with

771: 251:

The user may estimate the location of the decimal point in the result by mentally

4974: 4549:. "The difficulty of learning to use slide rules discouraged their use among the 4515: 4223: 3796: 3781: 3462: 3396: 3381: 3138: 3036: 2910: 2738: 91: 4932: 4888: 4210:

Ross, M. G. (2003). "Circle of time: errors in the use of the pregnancy wheel".

2329: 1047:

In addition to the logarithmic scales, some slide rules have other mathematical

4765: 4740: 4253: 3387: 3085: 2849: 2753: 2439: 2381: 67: 4715: 4489:

With the invention of the calculator the slide rule became instantly obsolete.

4379: 3809: 2358:

like the standard slide rule through the appropriate alignment of the scales.

43:

Typical ten-inch (25 cm) student slide rule (Pickett N902-T simplex trig)

5063: 4428: 4387: 4029: 3982: 3434: 3266: 3239: 2684: 2634: 2630: 2435: 2419: 2415: 2254: 165: 39: 3727:

Istvan, Szalkai (2016). "General Two-variable Functions on the Slide-rule".

2926:

Circular slide rules to estimate ovulation dates and fertility are known as

5029: 4672: 4505: 4231: 4045: 3560:

American Thesaurus of Slang: A Complete Reference Book of Colloquial Speech

3437: – Auxiliary scale of a measurement device, used to increase precision 2403: 2393: 2158: 1531:

and (2) to accurately determine cosine for small angles (with the S scale)

252: 209: 87: 3788:. An instruction manual for the unit marketed by Dietzgen can be found at 3567: 2811: 2333:

A 7-foot (2.1 m) teaching slide rule compared to a normal-sized model

1545: 3752: 3420: 3348:

Company collection, from the slide rule manufacturer formerly located in

3286: 3277:

particularly problematic, because the end-pieces on the cursors, made of

2974: 2894: 2853: 2842: 2793: 2773: 2742: 2676: 1552: 755:

for example, by positioning the top scale to start at the bottom scale's

233: 201: 185: 142: 130: 126: 4324: 4189: 4128:

Mathematics And Computer Education, Vol. 38, Iss. 1 (Winter 2004): 37–43

4037: 3997: 291: 5047: 4998: 4826: 4550: 3329: 3322: 3305: 2785:

it was used in far more limited numbers than conventional slide rules.

5014: 4160: 3974: 3615: 3584:

An Engineer's Alphabet: Gleanings from the Softer Side of a Profession

3289:

may be used to recreate missing or irretrievably broken cursor parts.

1978:. Inverse trigonometric functions are found by reversing the process. 935: 4273: 3785: 3294: 3285:

may be used to slow the deterioration of some older slide rules, and

3278: 2988:

Cryptographic slide rule used by the Swiss Army between 1914 and 1940

2861: 2857: 2680: 2667: 2623: 2548: 1038: 241: 83: 75: 3863:

Encyclopedia of the Scientific Revolution: From Copernicus to Newton

3207:. Statements consisting only of original research should be removed. 2901:

was a two-slide boxwood, brass, and cardboard device for estimating

267: 157:

and became inexpensive in the mid-1970s, slide rules became largely

4693: 3733: 3704: 3440: 3402: 3337: 3039: 2411: 2184: 513:

corresponds to shifting the top logarithmic scale by a distance of

158: 99: 79: 4956: 2622:

In 1895, a Japanese firm, Hemmi, started to make slide rules from

3090: 2835: 1876:), and then aligning the index ("1") of the C scale to the value 4985: 141:, it was the most commonly used calculation tool in science and 3496: 3485:

The slide rule is an example of a mechanical analog computer...

3411: – Mathematical instrument consisting of two hinged rulers 3369: 3080: 3047: 1975: 721:, the mark on the bottom scale at that position corresponds to 107: 5038:— French Slide Rules (Graphoplex, Tavernier-Gravet and others) 2860:

was sold at auction in 2007. The model N600-ES taken along on

2710:, expanding the slide rule's use beyond mathematical inquiry. 2376:

One slide rule remaining in daily use around the world is the

889:

mark are proportional to the logarithms of the marked values.

3074: 3063: 3055: 3021: 2645: 1433:

used along with the C and D scales for finding natural (base

154: 134: 103: 63: 3016: 2675:

The slide rule was invented around 1620–1630, shortly after

2242:{\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}} 4979: 4087:. Smithsonian National Air and Space Museum. Archived from 3152:

Being entirely mechanical, a slide rule does not depend on

2807: 2656: 2341:

Typically the divisions mark a scale to a precision of two

1056: 763:

can then be read on the bottom scale under the top scale's

52: 4982:— Dedicated to the preservation and history of slide rules 3907:

Kells, Lyman M.; Kern, Willis F.; Bland, James R. (1943).

3298:

2018. Proportion wheels are still used in graphic design.

3269:'s high-end models are the most popular among collectors. 3024:

scientific calculator, introduced for under US$ 25 in 1976

4962:

The history, theory and use of the engineering slide rule

4564:"Problem-based learning and the three C's of technology," 3910:

The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual

3161: 3105:

A slide rule requires the user to separately compute the

2920: 2377: 1757:

can be solved by first reducing the equation to the form

1071:) (for taking the log of a value on a multiplier scale), 3521:

For example, slide rules are mechanical analog computers

2923:

is a circular slide rule used by pilots and navigators.

2823:

not slack. After months of labour filling perhaps fifty

1051:

encoded on other auxiliary scales. The most popular are

4988:– Comprehensive Aristo site with many search facilities 4967: 4858:. Massachusetts Institute of Technology. Archived from 3928:

The Polyphase Duplex Slide Rule, A Self-Teaching Manual

3392:

Pages displaying short descriptions of redirect targets

3309:

event that the modern electronics in general use fail.

2373:

linear slide rules, and did not find reason to switch.

2362: 1137: 897:

The illustration below demonstrates the computation of

3810:"336 (Teknisk Tidskrift / 1933. Allmänna avdelningen)" 3775:

http://www.sphere.bc.ca/test/gilson/gilson-manual2.jpg

4673:"Greg's Slide Rules – Links to Slide Rule Collectors" 4511:

Beyond calculation: the next fifty years of computing

3790:

http://www.sliderulemuseum.com/SR_Library_General.htm

2263: 2202: 2020: 1943: 1902: 1882: 1848: 1811: 1763: 1712: 1661: 1618: 1588: 1496: 1459: 1439: 1401: 1371: 1270: 1234: 1214: 822:

For an example with even larger numbers, to multiply

727: 653: 603: 571: 551: 519: 499: 412: 335: 5054:

Descriptions, alphabetical by brandname, with images

3425:

Pages displaying wikidata descriptions as a fallback

3160:

computers, and not general-purpose calculators. The

2848:

Aluminium Pickett-brand slide rules were carried on

1151:

scales running from right to left, used to simplify

826:, the top scale is again positioned to start at the 4833:. Massachusetts Institute of Technology. 2005-01-11 2422:and gallon—liter fuel amount conversion functions. 2257:, etc.), and quadratic scales can be used to solve 2170:Using a linear L scale (available on some models): 4994:— Javascript simulations of historical slide rules 4909:Turner, Kimberli (2009-10-07). "Slide Rule: 101". 3688:"General Two-Variable Functions on the Slide Rule" 3557: 2834:designed and constructed a circular slide rule in 2302: 2241: 2143: 1958: 1916:. These two values are the roots of the equation. 1908: 1888: 1868: 1834: 1797: 1749: 1682: 1631: 1601: 1519: 1472: 1445: 1414: 1387: 1276: 1252: 1220: 739: 713: 639: 589: 557: 537: 505: 478: 400:{\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} 399: 149:were gradually introduced. But after the handheld 4310: 3913:. Keuffel & Esser. p. 92. Archived from 1986:models use decimal fractions of degrees instead. 5061: 4085:"Slide Rule, 5-inch, Pickett N600-ES, Apollo 13" 3558:Berrey, Lester V.; van den Bark, Melvin (1953). 3497:Ernst Bleuler; Robert Ozias Haxby (2011-09-21). 1647:on the C scale and go down to the LL scale with 1083:) scales. Others feature scales for calculating 858:, the answer must additionally be multiplied by 5050:— Information from The Museum of HP Calculators 4694:"Restoring and Repairing Slide Rules and Notes" 4500: 4212:Journal of Maternal-Fetal and Neonatal Medicine 3613: 3490: 3390: – Computer approximation for real numbers 3336:, has a collection of hundreds of slide rules, 1698: 1651:on it. That scale will indicate the answer. If 1132:versions of the C and D scales that start from 714:{\displaystyle \log(x)+\log(y)=\log(x\times y)} 66:for evaluating mathematical operations such as 4964:— By Dr James B. Calvert, University of Denver 4122:Sher, David B.; Nataro, Dean C. (2009-06-02). 3906: 3895:The Practical Mechanic and Engineer's Magazine 3616:"Eric's Types of Slide Rules and their Scales" 3587:. Cambridge University Press. pp. 46–47. 3460: 3321:A classroom-sized slide rule displayed at the 2905:from time of day, time of year, and latitude. 2872:five-inch pocket slide rule around with them. 2157:in specialized computer applications like the 1989: 838:, all numbers in that scale are multiplied by 811:. Thus, even though the answer directly reads 220:For example, a number to be multiplied on one 4606: 4423:Behrens, Lawrence; Rosen, Leonard J. (1982). 4340:"The HP 9100 Project: An Exothermic Reaction" 3102:(RPN) implemented in electronic calculators. 3012:History of computing hardware (1960s–present) 2507:The front side of a Boykin RotaRule Model 510 923:on the bottom scale. The resulting quotient, 121:English mathematician and clergyman Reverend 4813:"Commons category MIT Slide Rule Collection" 4686: 4422: 4331: 4304: 3946:. London: William Heinemann. pp. 76–78. 3384: – Circular slide rule used in aviation 3089:December 1951 advertisement for the IBM 604 3081:Comparison to electronic digital calculators 2519:The rear side of a Boykin RotaRule Model 510 479:{\displaystyle \log(x/y)=\log(x)-\log(y)\,.} 4352: 3281:, tend to chemically break down over time. 2852:space missions. The model N600-ES owned by 1998: 597:with the bottom scale's number at position 161:, so most suppliers departed the business. 4968:United Kingdom Slide Rule Circle Home Page 4754: 4729: 4449:. Princeton University Press. p. 16. 4121: 3751:. Tbullock.com. 2009-12-08. Archived from 2884: 2538: 168:, the slide rule is colloquially called a 4472: 4425:Writing and reading across the curriculum 4112:, American Technical Society, 1961, p. 5. 3856: 3732: 3703: 3526: 3223:Learn how and when to remove this message 2128: 2070: 927:, can then be read below the top scale's 472: 393: 275:Most slide rules consist of three parts: 4612: 4595:Stoll, Cliff. "When Slide Rules Ruled", 4578:Stoll, Cliff. "When Slide Rules Ruled," 4541:Stoll, Cliff. "When Slide Rules Ruled," 3580: 3454: 3316: 3234: 3084: 3015: 2958:John Rabone & Sons 1892 cattle gauge 2792:In the 1920s, the novelist and engineer 2752: 2666: 2655: 2466:A two-scale slide rule built into a ring 2328: 2324: 2183: 1919: 1037: 266: 179: 38: 4819: 4337: 3956: 3574: 3540:. Massachusetts Institute of Technology 3467:. Vol. 1. Macmillan. p. 175. 2663:(1575–1660), inventor of the slide rule 2352: 129:. It made calculations faster and less 14: 5062: 4908: 4358: 3939: 3726: 3685: 1564:The scales on the front and back of a 1014: 545:. This aligns each top scale's number 5032:— A very big Faber Castell collection 4980:Oughtred Society Slide Rule Home Page 4873: 4615:"In Praise Of RPN (with Python Or C)" 3995: 3829: 2679:'s publication of the concept of the 1197:on the C and CI (or D and DI) scales 318: 4844: 4444: 4209: 3921: 3900: 3836:. Courier Corporation. p. 205. 3172: 919:on the top scale is placed over the 4675:. Sliderule.ozmanor.com. 2004-07-29 4567:The Power of Problem-Based Learning 4062:. Heritage Auctions. Archived from 3897:, April 1845, p. 187 and Plate XX-B 3405: – Analog graphical calculator 3399: – American computer scientist 3250:A popular collectible model is the 3168: 2845:is that of the medical profession. 2691:of Cambridge combined two handheld 2495:wristwatch with circular slide rule 1577: 1284:) under the cursor on the DF scale 759:, the result of the multiplication 24: 4992:Derek's virtual slide rule gallery 3786:http://www.hpmuseum.org/srcirc.htm 2856:that flew with him to the Moon on 2313: 2179: 1208:– setting the cursor to any value 25: 5116: 5080:Historical scientific instruments 4950: 4475:Inventions that Changed the World 4368:Journal of VLSI Signal Processing 4022:10.1038/scientificamerican0506-80 3893:"Cameron's Nautical Slide Rule", 3071:University Interscholastic League 2671:1763 illustration of a slide rule 2303:{\displaystyle x^{2}+y^{2}=z^{2}} 1520:{\displaystyle {\sqrt {1-x^{2}}}} 488: 226:law of the logarithm of a product 175: 4254:"The Effects of Nuclear Weapons" 3857:Applebaum, Wilbur (2003-12-16). 3177: 2993: 2981: 2963: 2951: 2939: 2820:Fuller's cylindrical slide rules 2745:of the sun and principal stars. 2596: 2584: 2572: 2560: 2524: 2512: 2500: 2483: 2471: 2459: 2447: 2427: 1551: 1544: 934: 846:set of numbers is multiplied by 788: 770: 4957:International Slide Rule Museum 4929:International Slide Rule Museum 4917: 4902: 4805: 4789:. Faber-Castell. Archived from 4779: 4714:. Concise.co.jp. Archived from 4704: 4698:International Slide Rule Museum 4665: 4636: 4625:from the original on 2023-09-23 4589: 4572: 4556: 4535: 4494: 4466: 4438: 4416: 4286: 4266: 4246: 4203: 4174: 4145: 4115: 4102: 4077: 4052: 3989: 3950: 3933: 3887: 3850: 3823: 3802: 3799:. All retrieved March 14, 2007. 3766: 3741: 3720: 3692:Journal of the Oughtred Society 3503:. Academic Press. p. 638. 3357:International Slide Rule Museum 3133:of a slide rule is about three 3069:1980 was the final year of the 2188:Quadratic and reciprocal scales 1928: 1027: 640:{\displaystyle \log(x)+\log(y)} 5006:New International Encyclopedia 4925:"Slide Rule Library Main Desk" 4887:. Proun Design. Archived from 3679: 3669:"AntiQuark: Slide Rule Tricks" 3661: 3646:. pp. 7–8. Archived from 3632: 3614:Marcotte, Ph.D., Eric (2002). 3607: 3551: 3312: 2748: 1034:Slide rule scale § Scales 708: 696: 684: 678: 666: 660: 634: 628: 616: 610: 584: 578: 532: 526: 469: 463: 451: 445: 433: 419: 390: 384: 372: 366: 354: 342: 13: 1: 4359:Volder, Jack E. (June 2000). 4141:– via Findarticles.com. 3957:Witcher, C. M. (1954-12-01). 3447: 3417: – Mechanical calculator 3283:Methods of preserving plastic 2866:National Air and Space Museum 2806:on the design of the British 2796:(he called his autobiography 2604: 2531:Sperry 4016 Pocket Calculator 2402:In 1952, Swiss watch company 1750:{\displaystyle ax^{2}+bx+c=0} 262: 255:between labeled graduations. 114:is used to keep track of the 4911:Colorado Hometown Newspapers 4224:10.1080/14767050412331312200 3940:Norway, Nevil Shute (1954). 3930:, Breckenridge, 1922, p. 20. 3461:Roger R. Flynn (June 2002). 3261:) and a five-inch "pocket" ( 3091:Electronic Calculating Punch 2731:American Practical Navigator 2699:, with his one-time student 2614: 2192:Using (almost) any strictly 1798:{\displaystyle x^{2}-px+q=0} 1699:Roots of quadratic equations 1568:(K&E) 4181-3 slide rule 862:. The answer directly reads 854:in the top scale represents 313: 184:Simple slide rule made from 139:scientific pocket calculator 90:. It is one of the simplest 7: 4764:. ThinkGeek. Archived from 4739:. ThinkGeek. Archived from 4613:Williams, Al (2023-06-21). 4182:"Pilot Balloon Slide Rules" 3363: 3203:the claims made and adding 3119: × 0.03 (which equals 1990:Logarithms and exponentials 1655:is "off the scale", locate 1253:{\displaystyle {\pi }r^{2}} 892: 830:on the bottom scale. Since 803:set of numbers that number 137:. Before the advent of the 98:rule is closely related to 32:Slide rule (disambiguation) 10: 5121: 5036:Collection of slide rules 4881:"MIT Museum — Slide Rules" 4473:Castleden, Rodney (2007). 3009: 3005: 2651: 2317: 2161:(GRAPE) supercomputer and 1966:can be read directly from 1031: 989:, can be read beneath the 874:to get the actual answer: 842:. Thus, any answer in the 298: 29: 27:Mechanical analog computer 3784:. A photo can be seen at 3562:(2nd ed.). Crowell. 2946:An E6-B aviation computer 2155:logarithmic number system 1097:single-decade logarithmic 740:{\displaystyle x\times y} 4986:Rod Lovett's Slide Rules 4827:"MIT Museum measures up" 4447:e: The Story of a Number 4338:Leibson, Steven (2010). 4274:"Strangelove Slide Rule" 4108:Charles Overton Harris, 3998:"When Slide Rules Ruled" 3830:Smith, David E. (1958). 3686:Istvan, Szalkai (2016). 3581:Petroski, Henry (2011). 3372: – Calculating tool 3334:Cambridge, Massachusetts 3145:per kilogram). Multiple 2864:in 1970 is owned by the 2398:are still made and used. 1999:Addition and subtraction 1119:three-decade logarithmic 815:, the correct answer is 5105:17th-century inventions 4477:. Futura. p. 157. 4380:10.1023/a:1008110704586 4298:oldcalculatormuseum.com 3959:"Physics without sight" 3100:reverse Polish notation 3044:trigonometric functions 2885:Specialized calculators 2567:Fuller calculator, 1928 2539:Cylindrical slide rules 1959:{\displaystyle k\sin x} 1683:{\displaystyle x^{y/2}} 590:{\displaystyle \log(y)} 538:{\displaystyle \log(x)} 246:trigonometric functions 5090:Mechanical calculators 5056:(Vintage Tech. Assoc.) 5042:Eric's Slide Rule Site 5022:Encyclopedia Americana 3833:History of Mathematics 3342:mechanical calculators 3325: 3243: 3094: 3025: 2758: 2672: 2664: 2334: 2304: 2243: 2189: 2145: 1960: 1910: 1890: 1870: 1836: 1835:{\displaystyle p=-b/a} 1799: 1751: 1684: 1633: 1603: 1521: 1474: 1447: 1416: 1389: 1388:{\displaystyle e^{-x}} 1360:LL0N (or LL/N) and LLN 1344:on the C (or D) scale 1332:on the C (or D) scale 1320:on the C (or D) scale 1278: 1254: 1222: 1181:on the C (or D) scale 1108:two-decade logarithmic 1044: 741: 715: 641: 591: 559: 539: 507: 480: 401: 325:logarithmic identities 272: 271:Cursor on a slide rule 206:mathematical functions 189: 44: 4361:"The Birth of CORDIC" 4153:"Fisher sizing rules" 4110:Slide rule simplified 3996:Stoll, Cliff (2006). 3430:Timeline of computing 3376:Computer (occupation) 3320: 3242:slide rule with pouch 3238: 3088: 3073:(UIL) competition in 3019: 3000:Rare fractional adder 2779:Keuffel and Esser Co. 2756: 2708:Coggeshall slide rule 2670: 2659: 2332: 2325:Standard linear rules 2318:Further information: 2305: 2244: 2187: 2146: 1961: 1920:Future value of money 1911: 1891: 1871: 1869:{\displaystyle q=c/a} 1837: 1800: 1752: 1685: 1634: 1632:{\displaystyle x^{y}} 1604: 1602:{\displaystyle x^{2}} 1522: 1475: 1473:{\displaystyle e^{x}} 1448: 1417: 1415:{\displaystyle e^{x}} 1390: 1279: 1255: 1223: 1041: 966:would be computed as 785:is on the top scale: 742: 716: 642: 592: 560: 540: 508: 481: 402: 270: 183: 151:scientific calculator 57:mechanical calculator 42: 4580:Scientific American, 4543:Scientific American, 4313:Analytical Chemistry 4186:www.pilotballoon.com 3640:"instruction manual" 3274:online auction sites 3131:arithmetic precision 2971:Hurter and Driffield 2899:Hurter and Driffield 2603:Thacher slide rule, 2353:Circular slide rules 2261: 2251:parallel resistances 2200: 2163:hidden Markov models 2018: 1941: 1900: 1880: 1846: 1809: 1761: 1710: 1659: 1616: 1586: 1494: 1457: 1437: 1399: 1369: 1268: 1232: 1212: 1085:hyperbolic functions 725: 651: 601: 569: 549: 517: 497: 410: 333: 147:electronic computers 30:For other uses, see 4597:Scientific American 4506:Metcalfe, Robert M. 4325:10.1021/ac50155a005 4014:2006SciAm.294e..80S 4002:Scientific American 3871:2000esrc.book.....A 3714:2016arXiv161203955S 3409:Sector (instrument) 3350:Hoboken, New Jersey 3252:Keuffel & Esser 3147:speed-time-distance 2830:In 1937, physicist 2728:, described in the 2641:bearing surfaces. 2492:Breitling Navitimer 2442:conversion factors. 2408:Breitling Navitimer 2343:significant figures 2131: (subtraction) 1704:Quadratic equations 1529:Pythagorean theorem 1342:hyperbolic tangents 1015:Solving Proportions 257:Scientific notation 204:outputs of various 133:than evaluating on 112:scientific notation 5100:1622 introductions 5095:1620 introductions 5075:English inventions 5016:"Slide-rule" 5000:"Slide rule" 4973:2015-09-28 at the 4562:Watson, George H. 4445:Maor, Eli (2009). 3795:2007-02-14 at the 3780:2006-12-30 at the 3500:Electronic Methods 3326: 3244: 3188:possibly contains 3135:significant digits 3107:order of magnitude 3095: 3026: 2794:Nevil Shute Norway 2772:The growth of the 2759: 2726:Nathaniel Bowditch 2673: 2665: 2591:Bygrave slide rule 2553:Bygrave slide rule 2335: 2300: 2239: 2190: 2141: 2139: 1956: 1906: 1886: 1866: 1832: 1795: 1747: 1680: 1629: 1599: 1517: 1470: 1443: 1412: 1385: 1330:hyperbolic cosines 1274: 1250: 1228:on R1 or R2, find 1218: 1206:square root scales 1045: 737: 711: 637: 587: 555: 535: 503: 476: 397: 319:Logarithmic scales 306:may be grouped in 273: 190: 155:introduced in 1972 116:order of magnitude 45: 5044:— History and use 5030:Reglas de Cálculo 4525:978-0-387-98588-6 4502:Denning, Peter J. 4484:978-0-7088-0786-6 4456:978-0-691-14134-3 4294:"The Wang LOCI-2" 3975:10.1063/1.3061483 3510:978-0-08-085975-0 3474:978-0-02-865567-3 3464:Computer sciences 3346:Keuffel and Esser 3233: 3232: 3225: 3190:original research 3033:Wang Laboratories 2928:wheel calculators 2637:, sometimes with 2579:Otis King Model K 2545:Fuller calculator 2320:Slide rule scales 2237: 2224: 2211: 2132: 2096: 2074: 2038: 1909:{\displaystyle p} 1889:{\displaystyle q} 1572: 1571: 1566:Keuffel and Esser 1559: 1558: 1535: 1534: 1527:to (1) solve the 1515: 1453:) logarithms and 1446:{\displaystyle e} 1340:used for finding 1328:used for finding 1316:used for finding 1277:{\displaystyle r} 1221:{\displaystyle r} 1189:used for finding 1173:used for finding 1073:natural logarithm 807:be multiplied by 558:{\displaystyle y} 506:{\displaystyle x} 222:logarithmic-scale 16:(Redirected from 5112: 5070:Analog computers 5026: 5018: 5010: 5002: 4944: 4943: 4941: 4940: 4931:. Archived from 4921: 4915: 4914: 4906: 4900: 4899: 4897: 4896: 4877: 4871: 4870: 4868: 4867: 4848: 4842: 4841: 4839: 4838: 4823: 4817: 4816: 4809: 4803: 4801: 4799: 4798: 4787:"Rechenschieber" 4783: 4777: 4776: 4774: 4773: 4758: 4752: 4751: 4749: 4748: 4733: 4727: 4726: 4724: 4723: 4708: 4702: 4701: 4690: 4684: 4683: 4681: 4680: 4669: 4663: 4662: 4660: 4659: 4650:. Archived from 4648:citizenwatch.com 4640: 4634: 4633: 4631: 4630: 4610: 4604: 4593: 4587: 4576: 4570: 4560: 4554: 4539: 4533: 4532: 4498: 4492: 4491: 4470: 4464: 4463: 4442: 4436: 4435: 4420: 4414: 4413: 4411: 4410: 4404: 4398:. Archived from 4365: 4356: 4350: 4349: 4347: 4346: 4335: 4329: 4328: 4308: 4302: 4301: 4290: 4284: 4283: 4281: 4280: 4270: 4264: 4263: 4261: 4260: 4250: 4244: 4243: 4207: 4201: 4200: 4198: 4197: 4188:. Archived from 4178: 4172: 4171: 4169: 4168: 4159:. Archived from 4149: 4143: 4142: 4140: 4139: 4130:. Archived from 4119: 4113: 4106: 4100: 4099: 4097: 4096: 4081: 4075: 4074: 4072: 4071: 4056: 4050: 4049: 3993: 3987: 3986: 3954: 3948: 3947: 3937: 3931: 3925: 3919: 3918: 3904: 3898: 3891: 3885: 3884: 3854: 3848: 3847: 3827: 3821: 3820: 3818: 3817: 3806: 3800: 3770: 3764: 3763: 3761: 3760: 3745: 3739: 3738: 3736: 3724: 3718: 3717: 3707: 3683: 3677: 3676: 3665: 3659: 3658: 3656: 3655: 3636: 3630: 3629: 3627: 3626: 3620:www.sliderule.ca 3611: 3605: 3604: 3602: 3601: 3578: 3572: 3571: 3555: 3549: 3548: 3546: 3545: 3530: 3524: 3523: 3518: 3517: 3494: 3488: 3487: 3482: 3481: 3458: 3426: 3415:Slide calculator 3393: 3306:laying artillery 3228: 3221: 3217: 3214: 3208: 3205:inline citations 3181: 3180: 3173: 3169:Contemporary use 3154:grid electricity 3125: 3124: 3118: 3117: 3114: 2997: 2985: 2967: 2955: 2943: 2878:prosthaphaeresis 2804:Chief Calculator 2718:Peter Mark Roget 2701:Richard Delamain 2689:William Oughtred 2661:William Oughtred 2609: 2606: 2600: 2588: 2576: 2564: 2528: 2516: 2504: 2487: 2475: 2463: 2451: 2431: 2399: 2309: 2307: 2306: 2301: 2299: 2298: 2286: 2285: 2273: 2272: 2248: 2246: 2245: 2240: 2238: 2230: 2225: 2217: 2212: 2204: 2194:monotonic scales 2150: 2148: 2147: 2142: 2140: 2133: 2130: 2108: 2104: 2097: 2089: 2075: 2073: (addition) 2072: 2050: 2046: 2039: 2031: 1965: 1963: 1962: 1957: 1915: 1913: 1912: 1907: 1895: 1893: 1892: 1887: 1875: 1873: 1872: 1867: 1862: 1841: 1839: 1838: 1833: 1828: 1804: 1802: 1801: 1796: 1773: 1772: 1756: 1754: 1753: 1748: 1725: 1724: 1689: 1687: 1686: 1681: 1679: 1678: 1674: 1638: 1636: 1635: 1630: 1628: 1627: 1608: 1606: 1605: 1600: 1598: 1597: 1578:Roots and powers 1555: 1548: 1541: 1540: 1537: 1536: 1526: 1524: 1523: 1518: 1516: 1514: 1513: 1498: 1479: 1477: 1476: 1471: 1469: 1468: 1452: 1450: 1449: 1444: 1421: 1419: 1418: 1413: 1411: 1410: 1394: 1392: 1391: 1386: 1384: 1383: 1318:hyperbolic sines 1283: 1281: 1280: 1275: 1262:area of a circle 1259: 1257: 1256: 1251: 1249: 1248: 1239: 1227: 1225: 1224: 1219: 1164: 1163: 1159: 1145:CI, DI, CIF, DIF 1090: 1089: 1070: 1065:common logarithm 1010: 1009: 1007: 1006: 1003: 1000: 992: 988: 985:and the result, 984: 982: 980: 979: 976: 973: 965: 964: 962: 961: 958: 955: 944: 938: 930: 926: 922: 918: 914: 913: 911: 910: 907: 904: 888: 884: 881:In general, the 877: 873: 869: 865: 861: 857: 853: 849: 841: 837: 833: 829: 825: 818: 814: 810: 798: 792: 784: 780: 774: 766: 762: 758: 754: 750: 746: 744: 743: 738: 720: 718: 717: 712: 646: 644: 643: 638: 596: 594: 593: 588: 564: 562: 561: 556: 544: 542: 541: 536: 512: 510: 509: 504: 485: 483: 482: 477: 429: 406: 404: 403: 398: 123:William Oughtred 92:analog computers 21: 5120: 5119: 5115: 5114: 5113: 5111: 5110: 5109: 5060: 5059: 5013: 4997: 4975:Wayback Machine 4953: 4948: 4947: 4938: 4936: 4923: 4922: 4918: 4907: 4903: 4894: 4892: 4885:proundesign.com 4879: 4878: 4874: 4865: 4863: 4850: 4849: 4845: 4836: 4834: 4825: 4824: 4820: 4811: 4810: 4806: 4796: 4794: 4785: 4784: 4780: 4771: 4769: 4760: 4759: 4755: 4746: 4744: 4735: 4734: 4730: 4721: 4719: 4712:"About CONCISE" 4710: 4709: 4705: 4692: 4691: 4687: 4678: 4676: 4671: 4670: 4666: 4657: 4655: 4642: 4641: 4637: 4628: 4626: 4611: 4607: 4594: 4590: 4577: 4573: 4561: 4557: 4540: 4536: 4526: 4518:. p. xiv. 4499: 4495: 4485: 4471: 4467: 4457: 4443: 4439: 4431:. p. 273. 4421: 4417: 4408: 4406: 4402: 4363: 4357: 4353: 4344: 4342: 4336: 4332: 4319:(13): 62A–63A. 4309: 4305: 4292: 4291: 4287: 4278: 4276: 4272: 4271: 4267: 4258: 4256: 4252: 4251: 4247: 4208: 4204: 4195: 4193: 4180: 4179: 4175: 4166: 4164: 4151: 4150: 4146: 4137: 4135: 4120: 4116: 4107: 4103: 4094: 4092: 4083: 4082: 4078: 4069: 4067: 4058: 4057: 4053: 3994: 3990: 3955: 3951: 3938: 3934: 3926: 3922: 3905: 3901: 3892: 3888: 3881: 3855: 3851: 3844: 3828: 3824: 3815: 3813: 3808: 3807: 3803: 3797:Wayback Machine 3782:Wayback Machine 3771: 3767: 3758: 3756: 3747: 3746: 3742: 3725: 3721: 3684: 3680: 3667: 3666: 3662: 3653: 3651: 3638: 3637: 3633: 3624: 3622: 3612: 3608: 3599: 3597: 3595: 3579: 3575: 3556: 3552: 3543: 3541: 3532: 3531: 3527: 3515: 3513: 3511: 3495: 3491: 3479: 3477: 3475: 3459: 3455: 3450: 3424: 3397:Hans Peter Luhn 3391: 3382:Flight computer 3366: 3315: 3229: 3218: 3212: 3209: 3194: 3182: 3178: 3171: 3139:false precision 3122: 3120: 3115: 3112: 3110: 3083: 3037:Hewlett-Packard 3014: 3008: 3001: 2998: 2989: 2986: 2977: 2968: 2959: 2956: 2947: 2944: 2911:Fisher Controls 2887: 2818:(people) using 2763:Amédée Mannheim 2751: 2739:right ascension 2654: 2617: 2610: 2607: 2601: 2592: 2589: 2580: 2577: 2568: 2565: 2541: 2532: 2529: 2520: 2517: 2508: 2505: 2496: 2488: 2479: 2476: 2467: 2464: 2455: 2452: 2443: 2432: 2397: 2355: 2327: 2322: 2316: 2314:Physical design 2294: 2290: 2281: 2277: 2268: 2264: 2262: 2259: 2258: 2229: 2216: 2203: 2201: 2198: 2197: 2182: 2180:Generalizations 2138: 2137: 2129: 2112: 2088: 2087: 2083: 2080: 2079: 2071: 2054: 2030: 2029: 2025: 2021: 2019: 2016: 2015: 2001: 1992: 1942: 1939: 1938: 1931: 1922: 1901: 1898: 1897: 1881: 1878: 1877: 1858: 1847: 1844: 1843: 1824: 1810: 1807: 1806: 1768: 1764: 1762: 1759: 1758: 1720: 1716: 1711: 1708: 1707: 1701: 1670: 1666: 1662: 1660: 1657: 1656: 1623: 1619: 1617: 1614: 1613: 1593: 1589: 1587: 1584: 1583: 1580: 1509: 1505: 1497: 1495: 1492: 1491: 1464: 1460: 1458: 1455: 1454: 1438: 1435: 1434: 1406: 1402: 1400: 1397: 1396: 1376: 1372: 1370: 1367: 1366: 1269: 1266: 1265: 1244: 1240: 1235: 1233: 1230: 1229: 1213: 1210: 1209: 1161: 1157: 1156: 1068: 1036: 1030: 1017: 1004: 1001: 998: 997: 995: 994: 990: 986: 977: 974: 971: 970: 968: 967: 959: 956: 953: 952: 950: 949: 945:at either end. 942: 928: 924: 920: 916: 908: 905: 902: 901: 899: 898: 895: 886: 882: 875: 871: 867: 863: 859: 855: 851: 847: 839: 835: 831: 827: 823: 816: 812: 808: 796: 782: 778: 764: 760: 756: 752: 748: 726: 723: 722: 652: 649: 648: 602: 599: 598: 570: 567: 566: 550: 547: 546: 518: 515: 514: 498: 495: 494: 491: 425: 411: 408: 407: 334: 331: 330: 321: 316: 301: 265: 178: 35: 28: 23: 22: 15: 12: 11: 5: 5118: 5108: 5107: 5102: 5097: 5092: 5087: 5082: 5077: 5072: 5058: 5057: 5051: 5045: 5039: 5033: 5027: 5011: 4995: 4989: 4983: 4977: 4965: 4959: 4952: 4951:External links 4949: 4946: 4945: 4916: 4901: 4872: 4843: 4818: 4804: 4778: 4753: 4728: 4703: 4685: 4664: 4635: 4605: 4588: 4571: 4555: 4534: 4524: 4493: 4483: 4465: 4455: 4437: 4415: 4374:(2): 101–105. 4351: 4330: 4303: 4285: 4265: 4245: 4218:(6): 370–372. 4202: 4173: 4144: 4114: 4101: 4076: 4051: 3988: 3949: 3932: 3920: 3917:on 2009-02-14. 3899: 3886: 3879: 3849: 3842: 3822: 3812:. Runeberg.org 3801: 3765: 3740: 3719: 3678: 3660: 3631: 3606: 3593: 3573: 3550: 3525: 3509: 3489: 3473: 3452: 3451: 3449: 3446: 3445: 3444: 3438: 3432: 3427: 3418: 3412: 3406: 3400: 3394: 3388:Floating point 3385: 3379: 3373: 3365: 3362: 3314: 3311: 3231: 3230: 3185: 3183: 3176: 3170: 3167: 3082: 3079: 3007: 3004: 3003: 3002: 2999: 2992: 2990: 2987: 2980: 2978: 2969: 2962: 2960: 2957: 2950: 2948: 2945: 2938: 2886: 2883: 2862:Apollo 13 2858:Apollo 11 2850:Project Apollo 2750: 2747: 2653: 2650: 2616: 2613: 2612: 2611: 2602: 2595: 2593: 2590: 2583: 2581: 2578: 2571: 2569: 2566: 2559: 2540: 2537: 2534: 2533: 2530: 2523: 2521: 2518: 2511: 2509: 2506: 2499: 2497: 2489: 2482: 2480: 2477: 2470: 2468: 2465: 2458: 2456: 2453: 2446: 2444: 2433: 2426: 2382:dead reckoning 2354: 2351: 2326: 2323: 2315: 2312: 2297: 2293: 2289: 2284: 2280: 2276: 2271: 2267: 2236: 2233: 2228: 2223: 2220: 2215: 2210: 2207: 2181: 2178: 2177: 2176: 2175: 2174: 2168: 2167: 2166: 2151: 2136: 2127: 2124: 2121: 2118: 2115: 2113: 2111: 2107: 2103: 2100: 2095: 2092: 2086: 2082: 2081: 2078: 2069: 2066: 2063: 2060: 2057: 2055: 2053: 2049: 2045: 2042: 2037: 2034: 2028: 2024: 2023: 2000: 1997: 1991: 1988: 1955: 1952: 1949: 1946: 1930: 1927: 1921: 1918: 1905: 1885: 1865: 1861: 1857: 1854: 1851: 1831: 1827: 1823: 1820: 1817: 1814: 1794: 1791: 1788: 1785: 1782: 1779: 1776: 1771: 1767: 1746: 1743: 1740: 1737: 1734: 1731: 1728: 1723: 1719: 1715: 1700: 1697: 1677: 1673: 1669: 1665: 1626: 1622: 1596: 1592: 1579: 1576: 1570: 1569: 1561: 1560: 1557: 1556: 1549: 1533: 1532: 1512: 1508: 1504: 1501: 1485: 1481: 1480: 1467: 1463: 1442: 1428: 1424: 1423: 1409: 1405: 1382: 1379: 1375: 1364:log-log folded 1361: 1357: 1356: 1350: 1346: 1345: 1338: 1334: 1333: 1326: 1322: 1321: 1314: 1310: 1309: 1290: 1286: 1285: 1273: 1247: 1243: 1238: 1217: 1203: 1199: 1198: 1187: 1183: 1182: 1171: 1167: 1166: 1146: 1142: 1141: 1127: 1123: 1122: 1116: 1112: 1111: 1105: 1101: 1100: 1094: 1032:Main article: 1029: 1026: 1016: 1013: 894: 891: 866:. Multiply by 795:But since the 736: 733: 730: 710: 707: 704: 701: 698: 695: 692: 689: 686: 683: 680: 677: 674: 671: 668: 665: 662: 659: 656: 636: 633: 630: 627: 624: 621: 618: 615: 612: 609: 606: 586: 583: 580: 577: 574: 554: 534: 531: 528: 525: 522: 502: 490: 489:Multiplication 487: 475: 471: 468: 465: 462: 459: 456: 453: 450: 447: 444: 441: 438: 435: 432: 428: 424: 421: 418: 415: 396: 392: 389: 386: 383: 380: 377: 374: 371: 368: 365: 362: 359: 356: 353: 350: 347: 344: 341: 338: 323:The following 320: 317: 315: 312: 300: 297: 287: 286: 283: 280: 264: 261: 208:, acting as a 177: 176:Basic concepts 174: 68:multiplication 59:consisting of 26: 9: 6: 4: 3: 2: 5117: 5106: 5103: 5101: 5098: 5096: 5093: 5091: 5088: 5086: 5083: 5081: 5078: 5076: 5073: 5071: 5068: 5067: 5065: 5055: 5052: 5049: 5046: 5043: 5040: 5037: 5034: 5031: 5028: 5024: 5023: 5017: 5012: 5008: 5007: 5001: 4996: 4993: 4990: 4987: 4984: 4981: 4978: 4976: 4972: 4969: 4966: 4963: 4960: 4958: 4955: 4954: 4935:on 2023-01-07 4934: 4930: 4926: 4920: 4912: 4905: 4891:on 2019-05-01 4890: 4886: 4882: 4876: 4862:on 2019-05-01 4861: 4857: 4853: 4852:"Slide Rules" 4847: 4832: 4828: 4822: 4814: 4808: 4793:on 2013-11-21 4792: 4788: 4782: 4768:on 2012-04-15 4767: 4763: 4757: 4743:on 2010-03-27 4742: 4738: 4732: 4718:on 2012-03-12 4717: 4713: 4707: 4699: 4695: 4689: 4674: 4668: 4654:on 2014-04-22 4653: 4649: 4645: 4639: 4624: 4620: 4616: 4609: 4602: 4598: 4592: 4585: 4581: 4575: 4568: 4565: 4559: 4552: 4548: 4544: 4538: 4531: 4527: 4521: 4517: 4513: 4512: 4507: 4503: 4497: 4490: 4486: 4480: 4476: 4469: 4462: 4458: 4452: 4448: 4441: 4434: 4430: 4429:Little, Brown 4426: 4419: 4405:on 2016-03-04 4401: 4397: 4393: 4389: 4385: 4381: 4377: 4373: 4369: 4362: 4355: 4341: 4334: 4326: 4322: 4318: 4314: 4307: 4299: 4295: 4289: 4275: 4269: 4255: 4249: 4241: 4237: 4233: 4229: 4225: 4221: 4217: 4213: 4206: 4192:on 2016-09-28 4191: 4187: 4183: 4177: 4163:on 2010-01-06 4162: 4158: 4157:natgasedu.com 4154: 4148: 4134:on 2005-05-10 4133: 4129: 4125: 4118: 4111: 4105: 4091:on 2013-10-09 4090: 4086: 4080: 4066:on 2013-09-03 4065: 4061: 4055: 4047: 4043: 4039: 4035: 4031: 4027: 4023: 4019: 4015: 4011: 4007: 4003: 3999: 3992: 3984: 3980: 3976: 3972: 3968: 3964: 3963:Physics Today 3960: 3953: 3945: 3944: 3936: 3929: 3924: 3916: 3912: 3911: 3903: 3896: 3890: 3882: 3880:9781135582555 3876: 3872: 3868: 3865:. Routledge. 3864: 3860: 3853: 3845: 3843:9780486204307 3839: 3835: 3834: 3826: 3811: 3805: 3798: 3794: 3791: 3787: 3783: 3779: 3776: 3773:figures. 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