Slide rule - Knowledge

2418: 2515: 2984: 4575:. "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." 2587: 2972: 2438: 4592:. "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." 2462: 2474: 3307: 4558:, 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." 2563: 2942: 3087:

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

761: 3225: 779: 2319: 3075: 2491: 2503: 2358:

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.

29: 2551: 2450: 2354:). 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. 2754:, 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. 1535: 2657: 1542: 2954: 925: 4542:. 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." 2575: 2174: 257: 3168: 2930: 1129:) 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. 299:, 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. 1598:, 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. 3006: 2637:. 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. 2646: 170: 2758:

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.

2338:

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.

2417: 2743: 2138: 2357:

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

2326:

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

1913:

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

278:

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

3286:

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

3133:

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

2773:

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

1925:

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

1012:

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

3297:

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

3265:

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

2860:

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

2811:

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

2346:

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

1983:

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

3042:

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

2757:

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

2878:

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

2361:

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

1683:

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

86:

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

3761:

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

1926:

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

3086:

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

3153:

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

2337:

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

3148:

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

2377:

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.

2373:. 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 1679:

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.

2897:

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

2334:, 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. 2006: 4775: 283:

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.

3115:). 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). 3281:

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

2709:

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.

205:

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.

2615:-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. 1008:

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.

2777:

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.

3348:

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.

3039:(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. 2765:

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

3050:

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

2437: 1914:

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.

1031:

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

3235:

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.

2922:

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.

2544:, 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. 2385:. 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 1970:

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

2461: 2236: 1984:

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.

3138:

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.

937:

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

95:, 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 2011: 2702:

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.

394: 4779: 708: 1963:. 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/ 3047:

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.

473: 2514: 1028: 3017:

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.

930:

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

248:

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.

2297: 1514: 2133:{\displaystyle {\begin{aligned}\left({\frac {x}{y}}+1\right)y&=x+y\,{\text{ (addition)}},\\\left({\frac {x}{y}}-1\right)y&=x-y\,{\text{ (subtraction)}}.\end{aligned}}} 634: 2905:

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.

2870:, an algorithm for rapidly computing products that predates logarithms. However, there has been little practical interest in constructing one beyond the initial prototype. 2608:

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.

1744: 1792: 1088:

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

3066:

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.

1247: 177:

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.

1076:. 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. 734: 4632: 2723:

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.

3126:, 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 1953: 1677: 584: 532: 1829: 1382: 874:

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

2983: 2971: 1863: 1626: 1596: 1467: 1409: 766:

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

1903: 1883: 1440: 1271: 1215: 552: 500: 2676:

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,

770:

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

2423:

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

3897: 1684:

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.)

3341:, was donated to MIT around 2005, substantially expanding existing holdings. Selected items from the collection are usually on display at the museum. 3000: 4112: 2902:

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

3020:

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

4840: 3271: 2473: 134:. 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 4611: 4073: 3261:

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

778: 4450:

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

2327:

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

4349: 2692:

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

2443:

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

321: 5068: 4048: 3098:

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

214: 1885:

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

3766: 1110:

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

3931: 3847: 3781: 3571: 2786: 2586: 5042: 2350:

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

2188: 1992:

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

1099:

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

398: 3254:) variant. Another prized American model is the eight-inch (20 cm) Scientific Instruments circular rule. Of European rules, 2941: 4049:"Lot 25368 Buzz Aldrin's Apollo 11 Slide Rule – Flown to the Moon. ... 2007 September Grand Format Air & Space Auction #669" 2490: 3145:

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

2816:

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

2502: 4512: 4471: 4443: 3628: 3497: 3461: 2239: 2162:

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

4913: 4640: 2467:

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

4869: 1411:

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

4750: 4725: 1563:

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

4700: 2808: 2726:

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

2684:

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

2449: 3867: 3830: 3581: 3211: 3059: 3031:, introduced in 1968. Both of these were programmable and provided exponential and logarithmic functions; the HP had 2185:, other calculations can also be made with one movement. For example, reciprocal scales can be used for the equality 639: 4589: 4572: 4535: 3193: 1344:

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

5093: 4170: 3262: 1049: 5078: 4141: 3345: 271:

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

4791:

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

4422:

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

4300:

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

3947: 2953: 2395:

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

4994: 4552: 3178: 2830:

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

3903: 3737: 2618:

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

2854: 2864:

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

1922:

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

5024: 2719: 2562: 482:

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

2813: 2003:

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

5088: 5083: 5063: 4120: 3412: – hand-held device with moving parts, commonly paper, commonly printed for reference or calculation 135: 20: 3432: – Paper construct with rotating parts, may be considered a subclass of slide chart or movable book 2550: 1013:

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

274:

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

213:

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

4113:"The Prosthaphaeretic Slide Rule: A Mechanical Multiplication Device Based On Trigonometric Identities" 3246:, a premium scientific and engineering slide rule available both in a ten-inch (25 cm) "regular" ( 2249: 1482: 4848: 5058: 4603: 4077: 2374: 2143: 2142:

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

1571:

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

589: 3290:

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

2633:

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

2407:/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer— 114:

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

99:. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while 4052: 3322: 1698: 316:

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

221:

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

4959: 1749: 3189: 3088: 2627: 1996:

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

1220: 5030: 4988: 2803:

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

91:

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

5010: 4388: 3032: 2891: 2000: 1141: 1041: 1037: 982:

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

713: 313: 268:

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

234: 194: 60: 3820: 3676: 3487: 2929: 3418: 3364: 3330: 2767: 2696: 2399:. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured 1929: 1647: 557: 505: 218: 202: 139: 127: 45: 3035:(sine, cosine, and tangent) and hyperbolic trigonometric functions as well. The HP used the 2369:. This is a circular slide rule first created in the 1930s for aircraft pilots to help with 1797: 1357: 3998: 3855: 3698: 3119: 3024:

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

2959: 2887: 2751: 2574: 2532:

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

2151: 1834: 1604: 1574: 1445: 1387: 1073: 1065: 226: 5004: 3657: 3522: 3293:

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

1022: 8: 5073: 3763: 3397: 3338: 3306: 2695:

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

2685: 2480: 2396: 2331: 1692: 1517: 245: 100: 4002: 3859: 3778: 3702: 3367: – Person performing mathematical calculations, before calculators became available 2882:

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

2879:

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

4380: 4224: 4022: 3717: 3688: 3240: 3142: 3135: 3123: 3095: 2746:

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

2714: 2541: 2182: 1888: 1868: 1425: 1256: 1200: 537: 485: 296: 186: 104: 4519:

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

4010: 2381:

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

169: 4508: 4498: 4467: 4439: 4372: 4216: 4030: 4014: 3967: 3863: 3826: 3577: 3552: 3493: 3457: 3334: 3021: 2820: 2681: 2533: 2308: 1632:

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

1554: 1061: 210: 4980: 4801: 4228: 3185: 2750:

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

4490: 4364: 4309: 4282: 4208: 4006: 3959: 3403: 3082:

explicitly comparing electronic computers to engineers calculating with slide rules

2866: 2706: 2689: 2677: 2649: 1959:

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

1250: 1053: 292: 182: 111: 49: 4950: 4633:"Citizen Watch Company – Citizen Eco-Drive / US, Canada, UK, IrelandCitizen Watch" 4384: 4328: 3636: 3224: 3055:

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

1628:

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

760: 240:

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

4963: 4538:. "The difficulty of learning to use slide rules discouraged their use among the 4504: 4212: 3785: 3770: 3451: 3385: 3370: 3127: 3025: 2899: 2727: 80: 4921: 4877: 4199:

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

2318: 1036:

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

4754: 4729: 4242: 3376: 3074: 2838: 2742: 2428: 2370: 56: 4704: 4478:

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

4368: 3798: 2347:

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

32:

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

5052: 4417: 4376: 4018: 3971: 3423: 3255: 3228: 2673: 2623: 2619: 2424: 2408: 2404: 2243: 154: 28: 3716:

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

2915:

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

5018: 4661: 4494: 4220: 4034: 3549:

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

3426: – Auxiliary scale of a measurement device, used to increase precision 2392: 2382: 2147: 1520:

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

241: 198: 76: 3777:. An instruction manual for the unit marketed by Dietzgen can be found at 3556: 2800: 2322:

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

1534: 3741: 3409: 3337:

Company collection, from the slide rule manufacturer formerly located in

3275: 3266:

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

2963: 2883: 2842: 2831: 2782: 2762: 2731: 2665: 1541: 744:

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

222: 190: 174: 131: 119: 115: 4313: 4178: 4117:

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

4026: 3986: 280: 5036: 4987: 4815: 4539: 3318: 3311: 3294: 2774:

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

5003: 4149: 3963: 3604: 3573:

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

3278:

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

1967:. Inverse trigonometric functions are found by reversing the process. 924: 4262: 3774: 3283: 3274:

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

3267: 2977:

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

2850: 2846: 2669: 2656: 2612: 2537: 1027: 230: 72: 64: 3852:

Encyclopedia of the Scientific Revolution: From Copernicus to Newton

3196:. Statements consisting only of original research should be removed. 2890:

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

256: 146:

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

4682: 3722: 3693: 3429: 3391: 3326: 3028: 2400: 2173: 502:

corresponds to shifting the top logarithmic scale by a distance of

147: 88: 68: 4945: 2611:

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

3079: 2824: 1865:), and then aligning the index ("1") of the C scale to the value 4974: 130:, it was the most commonly used calculation tool in science and 3485: 3474:

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

3400: – Mathematical instrument consisting of two hinged rulers 3358: 3069: 3036: 1964: 710:, the mark on the bottom scale at that position corresponds to 96: 5027:— French Slide Rules (Graphoplex, Tavernier-Gravet and others) 2849:

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

2699:, expanding the slide rule's use beyond mathematical inquiry. 2365:

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

878:

mark are proportional to the logarithms of the marked values.

3063: 3052: 3044: 3010: 2634: 1422:

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

143: 123: 92: 52: 3005: 2664:

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

2231:{\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}} 4968: 4076:. Smithsonian National Air and Space Museum. Archived from 3141:

Being entirely mechanical, a slide rule does not depend on

2796: 2645: 2330:

Typically the divisions mark a scale to a precision of two

1045: 752:

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

41: 4971:— Dedicated to the preservation and history of slide rules 3896:

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

3287:

2018. Proportion wheels are still used in graphic design.

3258:'s high-end models are the most popular among collectors. 3013:

scientific calculator, introduced for under US$ 25 in 1976

4951:

The history, theory and use of the engineering slide rule

4553:"Problem-based learning and the three C's of technology," 3899:

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

3150: 3094:

A slide rule requires the user to separately compute the

2909: 2366: 1746:

can be solved by first reducing the equation to the form

1060:) (for taking the log of a value on a multiplier scale), 3510:

For example, slide rules are mechanical analog computers

2912:

is a circular slide rule used by pilots and navigators.

2812:

not slack. After months of labour filling perhaps fifty

1040:

encoded on other auxiliary scales. The most popular are

4977:– Comprehensive Aristo site with many search facilities 4956: 4847:. Massachusetts Institute of Technology. Archived from 3917:

The Polyphase Duplex Slide Rule, A Self-Teaching Manual

3381:

Pages displaying short descriptions of redirect targets

3298:

event that the modern electronics in general use fail.

2362:

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

2351: 1126: 886:

The illustration below demonstrates the computation of

3799:"336 (Teknisk Tidskrift / 1933. Allmänna avdelningen)" 3764:

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

4662:"Greg's Slide Rules – Links to Slide Rule Collectors" 4500:

Beyond calculation: the next fifty years of computing

3779:

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

2252: 2191: 2009: 1932: 1891: 1871: 1837: 1800: 1752: 1701: 1650: 1607: 1577: 1485: 1448: 1428: 1390: 1360: 1259: 1223: 1203: 811:

For an example with even larger numbers, to multiply

716: 642: 592: 560: 540: 508: 488: 401: 324: 5043:

Descriptions, alphabetical by brandname, with images

3414:

Pages displaying wikidata descriptions as a fallback

3149:

computers, and not general-purpose calculators. The

2837:

Aluminium Pickett-brand slide rules were carried on

1140:

scales running from right to left, used to simplify

815:, the top scale is again positioned to start at the 4822:. Massachusetts Institute of Technology. 2005-01-11 2411:and gallon—liter fuel amount conversion functions. 2246:, etc.), and quadratic scales can be used to solve 2159:Using a linear L scale (available on some models): 4983:— Javascript simulations of historical slide rules 4898:Turner, Kimberli (2009-10-07). "Slide Rule: 101". 3677:"General Two-Variable Functions on the Slide Rule" 3546: 2823:designed and constructed a circular slide rule in 2291: 2230: 2132: 1947: 1905:. These two values are the roots of the equation. 1897: 1877: 1857: 1823: 1786: 1738: 1671: 1620: 1590: 1508: 1461: 1434: 1403: 1376: 1265: 1241: 1209: 728: 702: 628: 578: 546: 526: 494: 467: 389:{\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} 388: 138:were gradually introduced. But after the handheld 4299: 3902:. Keuffel & Esser. p. 92. Archived from 1975:models use decimal fractions of degrees instead. 5050: 4074:"Slide Rule, 5-inch, Pickett N600-ES, Apollo 13" 3547:Berrey, Lester V.; van den Bark, Melvin (1953). 3486:Ernst Bleuler; Robert Ozias Haxby (2011-09-21). 1636:on the C scale and go down to the LL scale with 1072:) scales. Others feature scales for calculating 847:, the answer must additionally be multiplied by 5039:— Information from The Museum of HP Calculators 4683:"Restoring and Repairing Slide Rules and Notes" 4489: 4201:Journal of Maternal-Fetal and Neonatal Medicine 3602: 3479: 3379: – Computer approximation for real numbers 3325:, has a collection of hundreds of slide rules, 1687: 1640:on it. That scale will indicate the answer. If 1121:versions of the C and D scales that start from 703:{\displaystyle \log(x)+\log(y)=\log(x\times y)} 55:for evaluating mathematical operations such as 4953:— By Dr James B. Calvert, University of Denver 4111:Sher, David B.; Nataro, Dean C. (2009-06-02). 3895: 3884:The Practical Mechanic and Engineer's Magazine 3605:"Eric's Types of Slide Rules and their Scales" 3576:. Cambridge University Press. pp. 46–47. 3449: 3310:A classroom-sized slide rule displayed at the 2894:from time of day, time of year, and latitude. 2861:five-inch pocket slide rule around with them. 2146:in specialized computer applications like the 1978: 827:, all numbers in that scale are multiplied by 800:. Thus, even though the answer directly reads 209:For example, a number to be multiplied on one 4595: 4412:Behrens, Lawrence; Rosen, Leonard J. (1982). 4329:"The HP 9100 Project: An Exothermic Reaction" 3091:(RPN) implemented in electronic calculators. 3001:History of computing hardware (1960s–present) 2496:The front side of a Boykin RotaRule Model 510 912:on the bottom scale. The resulting quotient, 110:English mathematician and clergyman Reverend 4802:"Commons category MIT Slide Rule Collection" 4675: 4411: 4320: 4293: 3935:. London: William Heinemann. pp. 76–78. 3373: – Circular slide rule used in aviation 3078:December 1951 advertisement for the IBM 604 3070:Comparison to electronic digital calculators 2508:The rear side of a Boykin RotaRule Model 510 468:{\displaystyle \log(x/y)=\log(x)-\log(y)\,.} 4341: 3270:, tend to chemically break down over time. 2841:space missions. The model N600-ES owned by 1987: 586:with the bottom scale's number at position 150:, so most suppliers departed the business. 4957:United Kingdom Slide Rule Circle Home Page 4743: 4718: 4438:. Princeton University Press. p. 16. 4110: 3740:. Tbullock.com. 2009-12-08. Archived from 2873: 2527: 157:, the slide rule is colloquially called a 4461: 4414:Writing and reading across the curriculum 4101:, American Technical Society, 1961, p. 5. 3845: 3721: 3692: 3515: 3212:Learn how and when to remove this message 2117: 2059: 916:, can then be read below the top scale's 461: 382: 264:Most slide rules consist of three parts: 4601: 4584:Stoll, Cliff. "When Slide Rules Ruled", 4567:Stoll, Cliff. "When Slide Rules Ruled," 4530:Stoll, Cliff. "When Slide Rules Ruled," 3569: 3443: 3305: 3223: 3073: 3004: 2947:John Rabone & Sons 1892 cattle gauge 2781:In the 1920s, the novelist and engineer 2741: 2655: 2644: 2455:A two-scale slide rule built into a ring 2317: 2313: 2172: 1908: 1026: 255: 168: 27: 4808: 4326: 3945: 3563: 3529:. Massachusetts Institute of Technology 3456:. Vol. 1. Macmillan. p. 175. 2652:(1575–1660), inventor of the slide rule 2341: 118:. It made calculations faster and less 5051: 4897: 4347: 3928: 3715: 3674: 1553:The scales on the front and back of a 1003: 534:. This aligns each top scale's number 5021:— A very big Faber Castell collection 4969:Oughtred Society Slide Rule Home Page 4862: 4604:"In Praise Of RPN (with Python Or C)" 3984: 3818: 2668:'s publication of the concept of the 1186:on the C and CI (or D and DI) scales 307: 4833: 4433: 4198: 3910: 3889: 3825:. Courier Corporation. p. 205. 3161: 908:on the top scale is placed over the 4664:. Sliderule.ozmanor.com. 2004-07-29 4556:The Power of Problem-Based Learning 4051:. Heritage Auctions. Archived from 3886:, April 1845, p. 187 and Plate XX-B 3394: – Analog graphical calculator 3388: – American computer scientist 3239:A popular collectible model is the 3157: 2834:is that of the medical profession. 2680:of Cambridge combined two handheld 2484:wristwatch with circular slide rule 1566: 1273:) under the cursor on the DF scale 748:, the result of the multiplication 13: 4981:Derek's virtual slide rule gallery 3775:http://www.hpmuseum.org/srcirc.htm 2845:that flew with him to the Moon on 2302: 2168: 1197:– setting the cursor to any value 14: 5105: 5069:Historical scientific instruments 4939: 4464:Inventions that Changed the World 4357:Journal of VLSI Signal Processing 4011:10.1038/scientificamerican0506-80 3882:"Cameron's Nautical Slide Rule", 3060:University Interscholastic League 2660:1763 illustration of a slide rule 2292:{\displaystyle x^{2}+y^{2}=z^{2}} 1509:{\displaystyle {\sqrt {1-x^{2}}}} 477: 215:law of the logarithm of a product 164: 4243:"The Effects of Nuclear Weapons" 3846:Applebaum, Wilbur (2003-12-16). 3166: 2982: 2970: 2952: 2940: 2928: 2809:Fuller's cylindrical slide rules 2734:of the sun and principal stars. 2585: 2573: 2561: 2549: 2513: 2501: 2489: 2472: 2460: 2448: 2436: 2416: 1540: 1533: 923: 835:set of numbers is multiplied by 777: 759: 4946:International Slide Rule Museum 4918:International Slide Rule Museum 4906: 4891: 4794: 4778:. Faber-Castell. Archived from 4768: 4703:. Concise.co.jp. Archived from 4693: 4687:International Slide Rule Museum 4654: 4625: 4614:from the original on 2023-09-23 4578: 4561: 4545: 4524: 4483: 4455: 4427: 4405: 4275: 4255: 4235: 4192: 4163: 4134: 4104: 4091: 4066: 4041: 3978: 3939: 3922: 3876: 3839: 3812: 3791: 3788:. All retrieved March 14, 2007. 3755: 3730: 3709: 3681:Journal of the Oughtred Society 3492:. Academic Press. p. 638. 3346:International Slide Rule Museum 3122:of a slide rule is about three 3058:1980 was the final year of the 2177:Quadratic and reciprocal scales 1917: 1016: 629:{\displaystyle \log(x)+\log(y)} 4995:New International Encyclopedia 4914:"Slide Rule Library Main Desk" 4876:. Proun Design. Archived from 3668: 3658:"AntiQuark: Slide Rule Tricks" 3650: 3635:. pp. 7–8. Archived from 3621: 3603:Marcotte, Ph.D., Eric (2002). 3596: 3540: 3301: 2737: 1023:Slide rule scale § Scales 697: 685: 673: 667: 655: 649: 623: 617: 605: 599: 573: 567: 521: 515: 458: 452: 440: 434: 422: 408: 379: 373: 361: 355: 343: 331: 1: 4348:Volder, Jack E. (June 2000). 4130:– via Findarticles.com. 3946:Witcher, C. M. (1954-12-01). 3436: 3406: – Mechanical calculator 3272:Methods of preserving plastic 2855:National Air and Space Museum 2795:on the design of the British 2785:(he called his autobiography 2593: 2520:Sperry 4016 Pocket Calculator 2391:In 1952, Swiss watch company 1739:{\displaystyle ax^{2}+bx+c=0} 251: 244:between labeled graduations. 103:is used to keep track of the 4900:Colorado Hometown Newspapers 4213:10.1080/14767050412331312200 3929:Norway, Nevil Shute (1954). 3919:, Breckenridge, 1922, p. 20. 3450:Roger R. Flynn (June 2002). 3250:) and a five-inch "pocket" ( 3080:Electronic Calculating Punch 2720:American Practical Navigator 2688:, with his one-time student 2603: 2181:Using (almost) any strictly 1787:{\displaystyle x^{2}-px+q=0} 1688:Roots of quadratic equations 1557:(K&E) 4181-3 slide rule 851:. The answer directly reads 843:in the top scale represents 302: 173:Simple slide rule made from 128:scientific pocket calculator 79:. It is one of the simplest 7: 4753:. ThinkGeek. Archived from 4728:. ThinkGeek. Archived from 4602:Williams, Al (2023-06-21). 4171:"Pilot Balloon Slide Rules" 3352: 3192:the claims made and adding 3108: × 0.03 (which equals 1979:Logarithms and exponentials 1644:is "off the scale", locate 1242:{\displaystyle {\pi }r^{2}} 881: 819:on the bottom scale. Since 792:set of numbers that number 126:. Before the advent of the 87:rule is closely related to 21:Slide rule (disambiguation) 10: 5110: 5025:Collection of slide rules 4870:"MIT Museum — Slide Rules" 4462:Castleden, Rodney (2007). 2998: 2994: 2640: 2306: 2150:(GRAPE) supercomputer and 1955:can be read directly from 1020: 978:, can be read beneath the 863:to get the actual answer: 831:. Thus, any answer in the 287: 18: 16:Mechanical analog computer 3773:. A photo can be seen at 3551:(2nd ed.). Crowell. 2935:An E6-B aviation computer 2144:logarithmic number system 1086:single-decade logarithmic 729:{\displaystyle x\times y} 4975:Rod Lovett's Slide Rules 4816:"MIT Museum measures up" 4436:e: The Story of a Number 4327:Leibson, Steven (2010). 4263:"Strangelove Slide Rule" 4097:Charles Overton Harris, 3987:"When Slide Rules Ruled" 3819:Smith, David E. (1958). 3675:Istvan, Szalkai (2016). 3570:Petroski, Henry (2011). 3361: – Calculating tool 3323:Cambridge, Massachusetts 3134:per kilogram). Multiple 2853:in 1970 is owned by the 2387:are still made and used. 1988:Addition and subtraction 1108:three-decade logarithmic 804:, the correct answer is 5094:17th-century inventions 4466:. Futura. p. 157. 4369:10.1023/a:1008110704586 4287:oldcalculatormuseum.com 3948:"Physics without sight" 3089:reverse Polish notation 3033:trigonometric functions 2874:Specialized calculators 2556:Fuller calculator, 1928 2528:Cylindrical slide rules 1948:{\displaystyle k\sin x} 1672:{\displaystyle x^{y/2}} 579:{\displaystyle \log(y)} 527:{\displaystyle \log(x)} 235:trigonometric functions 5079:Mechanical calculators 5045:(Vintage Tech. Assoc.) 5031:Eric's Slide Rule Site 5011:Encyclopedia Americana 3822:History of Mathematics 3331:mechanical calculators 3314: 3232: 3083: 3014: 2747: 2661: 2653: 2323: 2293: 2232: 2178: 2134: 1949: 1899: 1879: 1859: 1825: 1824:{\displaystyle p=-b/a} 1788: 1740: 1673: 1622: 1592: 1510: 1463: 1436: 1405: 1378: 1377:{\displaystyle e^{-x}} 1349:LL0N (or LL/N) and LLN 1333:on the C (or D) scale 1321:on the C (or D) scale 1309:on the C (or D) scale 1267: 1243: 1211: 1170:on the C (or D) scale 1097:two-decade logarithmic 1033: 730: 704: 630: 580: 548: 528: 496: 469: 390: 314:logarithmic identities 261: 260:Cursor on a slide rule 195:mathematical functions 178: 33: 4350:"The Birth of CORDIC" 4142:"Fisher sizing rules" 4099:Slide rule simplified 3985:Stoll, Cliff (2006). 3419:Timeline of computing 3365:Computer (occupation) 3309: 3231:slide rule with pouch 3227: 3077: 3062:(UIL) competition in 3008: 2989:Rare fractional adder 2768:Keuffel and Esser Co. 2745: 2697:Coggeshall slide rule 2659: 2648: 2321: 2314:Standard linear rules 2307:Further information: 2294: 2233: 2176: 2135: 1950: 1909:Future value of money 1900: 1880: 1860: 1858:{\displaystyle q=c/a} 1826: 1789: 1741: 1674: 1623: 1621:{\displaystyle x^{y}} 1593: 1591:{\displaystyle x^{2}} 1511: 1464: 1462:{\displaystyle e^{x}} 1437: 1406: 1404:{\displaystyle e^{x}} 1379: 1268: 1244: 1212: 1030: 955:would be computed as 774:is on the top scale: 731: 705: 631: 581: 549: 529: 497: 470: 391: 259: 172: 140:scientific calculator 46:mechanical calculator 31: 4569:Scientific American, 4532:Scientific American, 4302:Analytical Chemistry 4175:www.pilotballoon.com 3629:"instruction manual" 3263:online auction sites 3120:arithmetic precision 2960:Hurter and Driffield 2888:Hurter and Driffield 2592:Thacher slide rule, 2342:Circular slide rules 2250: 2240:parallel resistances 2189: 2152:hidden Markov models 2007: 1930: 1889: 1869: 1835: 1798: 1750: 1699: 1648: 1605: 1575: 1483: 1446: 1426: 1388: 1358: 1257: 1221: 1201: 1074:hyperbolic functions 714: 640: 590: 558: 538: 506: 486: 399: 322: 136:electronic computers 19:For other uses, see 4586:Scientific American 4495:Metcalfe, Robert M. 4314:10.1021/ac50155a005 4003:2006SciAm.294e..80S 3991:Scientific American 3860:2000esrc.book.....A 3703:2016arXiv161203955S 3398:Sector (instrument) 3339:Hoboken, New Jersey 3241:Keuffel & Esser 3136:speed-time-distance 2819:In 1937, physicist 2717:, described in the 2630:bearing surfaces. 2481:Breitling Navitimer 2431:conversion factors. 2397:Breitling Navitimer 2332:significant figures 2120: (subtraction) 1693:Quadratic equations 1518:Pythagorean theorem 1331:hyperbolic tangents 1004:Solving Proportions 246:Scientific notation 193:outputs of various 122:than evaluating on 101:scientific notation 5089:1622 introductions 5084:1620 introductions 5064:English inventions 5005:"Slide-rule" 4989:"Slide rule" 4962:2015-09-28 at the 4551:Watson, George H. 4434:Maor, Eli (2009). 3784:2007-02-14 at the 3769:2006-12-30 at the 3489:Electronic Methods 3315: 3233: 3177:possibly contains 3124:significant digits 3096:order of magnitude 3084: 3015: 2783:Nevil Shute Norway 2761:The growth of the 2748: 2715:Nathaniel Bowditch 2662: 2654: 2580:Bygrave slide rule 2542:Bygrave slide rule 2324: 2289: 2228: 2179: 2130: 2128: 1945: 1895: 1875: 1855: 1821: 1784: 1736: 1669: 1618: 1588: 1506: 1459: 1432: 1401: 1374: 1319:hyperbolic cosines 1263: 1239: 1217:on R1 or R2, find 1207: 1195:square root scales 1034: 726: 700: 626: 576: 544: 524: 492: 465: 386: 308:Logarithmic scales 295:may be grouped in 262: 179: 144:introduced in 1972 105:order of magnitude 34: 5033:— History and use 5019:Reglas de Cálculo 4514:978-0-387-98588-6 4491:Denning, Peter J. 4473:978-0-7088-0786-6 4445:978-0-691-14134-3 4283:"The Wang LOCI-2" 3964:10.1063/1.3061483 3499:978-0-08-085975-0 3463:978-0-02-865567-3 3453:Computer sciences 3335:Keuffel and Esser 3222: 3221: 3214: 3179:original research 3022:Wang Laboratories 2917:wheel calculators 2626:, sometimes with 2568:Otis King Model K 2534:Fuller calculator 2309:Slide rule scales 2226: 2213: 2200: 2121: 2085: 2063: 2027: 1898:{\displaystyle p} 1878:{\displaystyle q} 1561: 1560: 1555:Keuffel and Esser 1548: 1547: 1524: 1523: 1516:to (1) solve the 1504: 1442:) logarithms and 1435:{\displaystyle e} 1329:used for finding 1317:used for finding 1305:used for finding 1266:{\displaystyle r} 1210:{\displaystyle r} 1178:used for finding 1162:used for finding 1062:natural logarithm 796:be multiplied by 547:{\displaystyle y} 495:{\displaystyle x} 211:logarithmic-scale 5101: 5059:Analog computers 5015: 5007: 4999: 4991: 4933: 4932: 4930: 4929: 4920:. Archived from 4910: 4904: 4903: 4895: 4889: 4888: 4886: 4885: 4866: 4860: 4859: 4857: 4856: 4837: 4831: 4830: 4828: 4827: 4812: 4806: 4805: 4798: 4792: 4790: 4788: 4787: 4776:"Rechenschieber" 4772: 4766: 4765: 4763: 4762: 4747: 4741: 4740: 4738: 4737: 4722: 4716: 4715: 4713: 4712: 4697: 4691: 4690: 4679: 4673: 4672: 4670: 4669: 4658: 4652: 4651: 4649: 4648: 4639:. Archived from 4637:citizenwatch.com 4629: 4623: 4622: 4620: 4619: 4599: 4593: 4582: 4576: 4565: 4559: 4549: 4543: 4528: 4522: 4521: 4487: 4481: 4480: 4459: 4453: 4452: 4431: 4425: 4424: 4409: 4403: 4402: 4400: 4399: 4393: 4387:. Archived from 4354: 4345: 4339: 4338: 4336: 4335: 4324: 4318: 4317: 4297: 4291: 4290: 4279: 4273: 4272: 4270: 4269: 4259: 4253: 4252: 4250: 4249: 4239: 4233: 4232: 4196: 4190: 4189: 4187: 4186: 4177:. Archived from 4167: 4161: 4160: 4158: 4157: 4148:. Archived from 4138: 4132: 4131: 4129: 4128: 4119:. Archived from 4108: 4102: 4095: 4089: 4088: 4086: 4085: 4070: 4064: 4063: 4061: 4060: 4045: 4039: 4038: 3982: 3976: 3975: 3943: 3937: 3936: 3926: 3920: 3914: 3908: 3907: 3893: 3887: 3880: 3874: 3873: 3843: 3837: 3836: 3816: 3810: 3809: 3807: 3806: 3795: 3789: 3759: 3753: 3752: 3750: 3749: 3734: 3728: 3727: 3725: 3713: 3707: 3706: 3696: 3672: 3666: 3665: 3654: 3648: 3647: 3645: 3644: 3625: 3619: 3618: 3616: 3615: 3609:www.sliderule.ca 3600: 3594: 3593: 3591: 3590: 3567: 3561: 3560: 3544: 3538: 3537: 3535: 3534: 3519: 3513: 3512: 3507: 3506: 3483: 3477: 3476: 3471: 3470: 3447: 3415: 3404:Slide calculator 3382: 3295:laying artillery 3217: 3210: 3206: 3203: 3197: 3194:inline citations 3170: 3169: 3162: 3158:Contemporary use 3143:grid electricity 3114: 3113: 3107: 3106: 3103: 2986: 2974: 2956: 2944: 2932: 2867:prosthaphaeresis 2793:Chief Calculator 2707:Peter Mark Roget 2690:Richard Delamain 2678:William Oughtred 2650:William Oughtred 2598: 2595: 2589: 2577: 2565: 2553: 2517: 2505: 2493: 2476: 2464: 2452: 2440: 2420: 2388: 2298: 2296: 2295: 2290: 2288: 2287: 2275: 2274: 2262: 2261: 2237: 2235: 2234: 2229: 2227: 2219: 2214: 2206: 2201: 2193: 2183:monotonic scales 2139: 2137: 2136: 2131: 2129: 2122: 2119: 2097: 2093: 2086: 2078: 2064: 2062: (addition) 2061: 2039: 2035: 2028: 2020: 1954: 1952: 1951: 1946: 1904: 1902: 1901: 1896: 1884: 1882: 1881: 1876: 1864: 1862: 1861: 1856: 1851: 1830: 1828: 1827: 1822: 1817: 1793: 1791: 1790: 1785: 1762: 1761: 1745: 1743: 1742: 1737: 1714: 1713: 1678: 1676: 1675: 1670: 1668: 1667: 1663: 1627: 1625: 1624: 1619: 1617: 1616: 1597: 1595: 1594: 1589: 1587: 1586: 1567:Roots and powers 1544: 1537: 1530: 1529: 1526: 1525: 1515: 1513: 1512: 1507: 1505: 1503: 1502: 1487: 1468: 1466: 1465: 1460: 1458: 1457: 1441: 1439: 1438: 1433: 1410: 1408: 1407: 1402: 1400: 1399: 1383: 1381: 1380: 1375: 1373: 1372: 1307:hyperbolic sines 1272: 1270: 1269: 1264: 1251:area of a circle 1248: 1246: 1245: 1240: 1238: 1237: 1228: 1216: 1214: 1213: 1208: 1153: 1152: 1148: 1134:CI, DI, CIF, DIF 1079: 1078: 1059: 1054:common logarithm 999: 998: 996: 995: 992: 989: 981: 977: 974:and the result, 973: 971: 969: 968: 965: 962: 954: 953: 951: 950: 947: 944: 933: 927: 919: 915: 911: 907: 903: 902: 900: 899: 896: 893: 877: 873: 870:In general, the 866: 862: 858: 854: 850: 846: 842: 838: 830: 826: 822: 818: 814: 807: 803: 799: 787: 781: 773: 769: 763: 755: 751: 747: 743: 739: 735: 733: 732: 727: 709: 707: 706: 701: 635: 633: 632: 627: 585: 583: 582: 577: 553: 551: 550: 545: 533: 531: 530: 525: 501: 499: 498: 493: 474: 472: 471: 466: 418: 395: 393: 392: 387: 112:William Oughtred 81:analog computers 5109: 5108: 5104: 5103: 5102: 5100: 5099: 5098: 5049: 5048: 5002: 4986: 4964:Wayback Machine 4942: 4937: 4936: 4927: 4925: 4912: 4911: 4907: 4896: 4892: 4883: 4881: 4874:proundesign.com 4868: 4867: 4863: 4854: 4852: 4839: 4838: 4834: 4825: 4823: 4814: 4813: 4809: 4800: 4799: 4795: 4785: 4783: 4774: 4773: 4769: 4760: 4758: 4749: 4748: 4744: 4735: 4733: 4724: 4723: 4719: 4710: 4708: 4701:"About CONCISE" 4699: 4698: 4694: 4681: 4680: 4676: 4667: 4665: 4660: 4659: 4655: 4646: 4644: 4631: 4630: 4626: 4617: 4615: 4600: 4596: 4583: 4579: 4566: 4562: 4550: 4546: 4529: 4525: 4515: 4507:. p. xiv. 4488: 4484: 4474: 4460: 4456: 4446: 4432: 4428: 4420:. p. 273. 4410: 4406: 4397: 4395: 4391: 4352: 4346: 4342: 4333: 4331: 4325: 4321: 4308:(13): 62A–63A. 4298: 4294: 4281: 4280: 4276: 4267: 4265: 4261: 4260: 4256: 4247: 4245: 4241: 4240: 4236: 4197: 4193: 4184: 4182: 4169: 4168: 4164: 4155: 4153: 4140: 4139: 4135: 4126: 4124: 4109: 4105: 4096: 4092: 4083: 4081: 4072: 4071: 4067: 4058: 4056: 4047: 4046: 4042: 3983: 3979: 3944: 3940: 3927: 3923: 3915: 3911: 3894: 3890: 3881: 3877: 3870: 3844: 3840: 3833: 3817: 3813: 3804: 3802: 3797: 3796: 3792: 3786:Wayback Machine 3771:Wayback Machine 3760: 3756: 3747: 3745: 3736: 3735: 3731: 3714: 3710: 3673: 3669: 3656: 3655: 3651: 3642: 3640: 3627: 3626: 3622: 3613: 3611: 3601: 3597: 3588: 3586: 3584: 3568: 3564: 3545: 3541: 3532: 3530: 3521: 3520: 3516: 3504: 3502: 3500: 3484: 3480: 3468: 3466: 3464: 3448: 3444: 3439: 3413: 3386:Hans Peter Luhn 3380: 3371:Flight computer 3355: 3304: 3218: 3207: 3201: 3198: 3183: 3171: 3167: 3160: 3128:false precision 3111: 3109: 3104: 3101: 3099: 3072: 3026:Hewlett-Packard 3003: 2997: 2990: 2987: 2978: 2975: 2966: 2957: 2948: 2945: 2936: 2933: 2900:Fisher Controls 2876: 2807:(people) using 2752:Amédée Mannheim 2740: 2728:right ascension 2643: 2606: 2599: 2596: 2590: 2581: 2578: 2569: 2566: 2557: 2554: 2530: 2521: 2518: 2509: 2506: 2497: 2494: 2485: 2477: 2468: 2465: 2456: 2453: 2444: 2441: 2432: 2421: 2386: 2344: 2316: 2311: 2305: 2303:Physical design 2283: 2279: 2270: 2266: 2257: 2253: 2251: 2248: 2247: 2218: 2205: 2192: 2190: 2187: 2186: 2171: 2169:Generalizations 2127: 2126: 2118: 2101: 2077: 2076: 2072: 2069: 2068: 2060: 2043: 2019: 2018: 2014: 2010: 2008: 2005: 2004: 1990: 1981: 1931: 1928: 1927: 1920: 1911: 1890: 1887: 1886: 1870: 1867: 1866: 1847: 1836: 1833: 1832: 1813: 1799: 1796: 1795: 1757: 1753: 1751: 1748: 1747: 1709: 1705: 1700: 1697: 1696: 1690: 1659: 1655: 1651: 1649: 1646: 1645: 1612: 1608: 1606: 1603: 1602: 1582: 1578: 1576: 1573: 1572: 1569: 1498: 1494: 1486: 1484: 1481: 1480: 1453: 1449: 1447: 1444: 1443: 1427: 1424: 1423: 1395: 1391: 1389: 1386: 1385: 1365: 1361: 1359: 1356: 1355: 1258: 1255: 1254: 1233: 1229: 1224: 1222: 1219: 1218: 1202: 1199: 1198: 1150: 1146: 1145: 1057: 1025: 1019: 1006: 993: 990: 987: 986: 984: 983: 979: 975: 966: 963: 960: 959: 957: 956: 948: 945: 942: 941: 939: 938: 934:at either end. 931: 917: 913: 909: 905: 897: 894: 891: 890: 888: 887: 884: 875: 871: 864: 860: 856: 852: 848: 844: 840: 836: 828: 824: 820: 816: 812: 805: 801: 797: 785: 771: 767: 753: 749: 745: 741: 737: 715: 712: 711: 641: 638: 637: 591: 588: 587: 559: 556: 555: 539: 536: 535: 507: 504: 503: 487: 484: 483: 480: 414: 400: 397: 396: 323: 320: 319: 310: 305: 290: 254: 167: 24: 17: 12: 11: 5: 5107: 5097: 5096: 5091: 5086: 5081: 5076: 5071: 5066: 5061: 5047: 5046: 5040: 5034: 5028: 5022: 5016: 5000: 4984: 4978: 4972: 4966: 4954: 4948: 4941: 4940:External links 4938: 4935: 4934: 4905: 4890: 4861: 4832: 4807: 4793: 4767: 4742: 4717: 4692: 4674: 4653: 4624: 4594: 4577: 4560: 4544: 4523: 4513: 4482: 4472: 4454: 4444: 4426: 4404: 4363:(2): 101–105. 4340: 4319: 4292: 4274: 4254: 4234: 4207:(6): 370–372. 4191: 4162: 4133: 4103: 4090: 4065: 4040: 3977: 3938: 3921: 3909: 3906:on 2009-02-14. 3888: 3875: 3868: 3838: 3831: 3811: 3801:. Runeberg.org 3790: 3754: 3729: 3708: 3667: 3649: 3620: 3595: 3582: 3562: 3539: 3514: 3498: 3478: 3462: 3441: 3440: 3438: 3435: 3434: 3433: 3427: 3421: 3416: 3407: 3401: 3395: 3389: 3383: 3377:Floating point 3374: 3368: 3362: 3354: 3351: 3303: 3300: 3220: 3219: 3174: 3172: 3165: 3159: 3156: 3071: 3068: 2996: 2993: 2992: 2991: 2988: 2981: 2979: 2976: 2969: 2967: 2958: 2951: 2949: 2946: 2939: 2937: 2934: 2927: 2875: 2872: 2851:Apollo 13 2847:Apollo 11 2839:Project Apollo 2739: 2736: 2642: 2639: 2605: 2602: 2601: 2600: 2591: 2584: 2582: 2579: 2572: 2570: 2567: 2560: 2558: 2555: 2548: 2529: 2526: 2523: 2522: 2519: 2512: 2510: 2507: 2500: 2498: 2495: 2488: 2486: 2478: 2471: 2469: 2466: 2459: 2457: 2454: 2447: 2445: 2442: 2435: 2433: 2422: 2415: 2371:dead reckoning 2343: 2340: 2315: 2312: 2304: 2301: 2286: 2282: 2278: 2273: 2269: 2265: 2260: 2256: 2225: 2222: 2217: 2212: 2209: 2204: 2199: 2196: 2170: 2167: 2166: 2165: 2164: 2163: 2157: 2156: 2155: 2140: 2125: 2116: 2113: 2110: 2107: 2104: 2102: 2100: 2096: 2092: 2089: 2084: 2081: 2075: 2071: 2070: 2067: 2058: 2055: 2052: 2049: 2046: 2044: 2042: 2038: 2034: 2031: 2026: 2023: 2017: 2013: 2012: 1989: 1986: 1980: 1977: 1944: 1941: 1938: 1935: 1919: 1916: 1910: 1907: 1894: 1874: 1854: 1850: 1846: 1843: 1840: 1820: 1816: 1812: 1809: 1806: 1803: 1783: 1780: 1777: 1774: 1771: 1768: 1765: 1760: 1756: 1735: 1732: 1729: 1726: 1723: 1720: 1717: 1712: 1708: 1704: 1689: 1686: 1666: 1662: 1658: 1654: 1615: 1611: 1585: 1581: 1568: 1565: 1559: 1558: 1550: 1549: 1546: 1545: 1538: 1522: 1521: 1501: 1497: 1493: 1490: 1474: 1470: 1469: 1456: 1452: 1431: 1417: 1413: 1412: 1398: 1394: 1371: 1368: 1364: 1353:log-log folded 1350: 1346: 1345: 1339: 1335: 1334: 1327: 1323: 1322: 1315: 1311: 1310: 1303: 1299: 1298: 1279: 1275: 1274: 1262: 1236: 1232: 1227: 1206: 1192: 1188: 1187: 1176: 1172: 1171: 1160: 1156: 1155: 1135: 1131: 1130: 1116: 1112: 1111: 1105: 1101: 1100: 1094: 1090: 1089: 1083: 1021:Main article: 1018: 1015: 1005: 1002: 883: 880: 855:. Multiply by 784:But since the 725: 722: 719: 699: 696: 693: 690: 687: 684: 681: 678: 675: 672: 669: 666: 663: 660: 657: 654: 651: 648: 645: 625: 622: 619: 616: 613: 610: 607: 604: 601: 598: 595: 575: 572: 569: 566: 563: 543: 523: 520: 517: 514: 511: 491: 479: 478:Multiplication 476: 464: 460: 457: 454: 451: 448: 445: 442: 439: 436: 433: 430: 427: 424: 421: 417: 413: 410: 407: 404: 385: 381: 378: 375: 372: 369: 366: 363: 360: 357: 354: 351: 348: 345: 342: 339: 336: 333: 330: 327: 312:The following 309: 306: 304: 301: 289: 286: 276: 275: 272: 269: 253: 250: 197:, acting as a 166: 165:Basic concepts 163: 57:multiplication 48:consisting of 15: 9: 6: 4: 3: 2: 5106: 5095: 5092: 5090: 5087: 5085: 5082: 5080: 5077: 5075: 5072: 5070: 5067: 5065: 5062: 5060: 5057: 5056: 5054: 5044: 5041: 5038: 5035: 5032: 5029: 5026: 5023: 5020: 5017: 5013: 5012: 5006: 5001: 4997: 4996: 4990: 4985: 4982: 4979: 4976: 4973: 4970: 4967: 4965: 4961: 4958: 4955: 4952: 4949: 4947: 4944: 4943: 4924:on 2023-01-07 4923: 4919: 4915: 4909: 4901: 4894: 4880:on 2019-05-01 4879: 4875: 4871: 4865: 4851:on 2019-05-01 4850: 4846: 4842: 4841:"Slide Rules" 4836: 4821: 4817: 4811: 4803: 4797: 4782:on 2013-11-21 4781: 4777: 4771: 4757:on 2012-04-15 4756: 4752: 4746: 4732:on 2010-03-27 4731: 4727: 4721: 4707:on 2012-03-12 4706: 4702: 4696: 4688: 4684: 4678: 4663: 4657: 4643:on 2014-04-22 4642: 4638: 4634: 4628: 4613: 4609: 4605: 4598: 4591: 4587: 4581: 4574: 4570: 4564: 4557: 4554: 4548: 4541: 4537: 4533: 4527: 4520: 4516: 4510: 4506: 4502: 4501: 4496: 4492: 4486: 4479: 4475: 4469: 4465: 4458: 4451: 4447: 4441: 4437: 4430: 4423: 4419: 4418:Little, Brown 4415: 4408: 4394:on 2016-03-04 4390: 4386: 4382: 4378: 4374: 4370: 4366: 4362: 4358: 4351: 4344: 4330: 4323: 4315: 4311: 4307: 4303: 4296: 4288: 4284: 4278: 4264: 4258: 4244: 4238: 4230: 4226: 4222: 4218: 4214: 4210: 4206: 4202: 4195: 4181:on 2016-09-28 4180: 4176: 4172: 4166: 4152:on 2010-01-06 4151: 4147: 4146:natgasedu.com 4143: 4137: 4123:on 2005-05-10 4122: 4118: 4114: 4107: 4100: 4094: 4080:on 2013-10-09 4079: 4075: 4069: 4055:on 2013-09-03 4054: 4050: 4044: 4036: 4032: 4028: 4024: 4020: 4016: 4012: 4008: 4004: 4000: 3996: 3992: 3988: 3981: 3973: 3969: 3965: 3961: 3957: 3953: 3952:Physics Today 3949: 3942: 3934: 3933: 3925: 3918: 3913: 3905: 3901: 3900: 3892: 3885: 3879: 3871: 3869:9781135582555 3865: 3861: 3857: 3854:. Routledge. 3853: 3849: 3842: 3834: 3832:9780486204307 3828: 3824: 3823: 3815: 3800: 3794: 3787: 3783: 3780: 3776: 3772: 3768: 3765: 3762:figures. See 3758: 3744:on 2013-02-03 3743: 3739: 3738:"Slide Rules" 3733: 3724: 3719: 3712: 3704: 3700: 3695: 3690: 3686: 3682: 3678: 3671: 3663: 3662:antiquark.com 3659: 3653: 3639:on 2007-04-02 3638: 3634: 3630: 3624: 3610: 3606: 3599: 3585: 3583:9781139505307 3579: 3575: 3574: 3566: 3558: 3554: 3550: 3543: 3528: 3524: 3523:"Slide Rules" 3518: 3511: 3501: 3495: 3491: 3490: 3482: 3475: 3465: 3459: 3455: 3454: 3446: 3442: 3431: 3428: 3425: 3424:Vernier scale 3422: 3420: 3417: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3390: 3387: 3384: 3378: 3375: 3372: 3369: 3366: 3363: 3360: 3357: 3356: 3350: 3347: 3342: 3340: 3336: 3332: 3328: 3324: 3320: 3313: 3308: 3299: 3296: 3291: 3288: 3285: 3279: 3277: 3273: 3269: 3264: 3259: 3257: 3256:Faber-Castell 3253: 3249: 3245: 3242: 3237: 3230: 3229:Faber-Castell 3226: 3216: 3213: 3205: 3202:February 2015 3195: 3191: 3187: 3181: 3180: 3175:This section 3173: 3164: 3163: 3155: 3152: 3146: 3144: 3139: 3137: 3131: 3129: 3125: 3121: 3116: 3097: 3092: 3090: 3081: 3076: 3067: 3065: 3061: 3056: 3054: 3048: 3046: 3040: 3038: 3034: 3030: 3027: 3023: 3018: 3012: 3007: 3002: 2985: 2980: 2973: 2968: 2965: 2961: 2955: 2950: 2943: 2938: 2931: 2926: 2925: 2924: 2920: 2918: 2913: 2911: 2906: 2903: 2901: 2895: 2893: 2889: 2885: 2880: 2871: 2869: 2868: 2862: 2858: 2856: 2852: 2848: 2844: 2840: 2835: 2833: 2828: 2826: 2822: 2817: 2815: 2810: 2806: 2802: 2798: 2794: 2790: 2789: 2784: 2779: 2775: 2771: 2770:of New York. 2769: 2764: 2759: 2755: 2753: 2744: 2735: 2733: 2729: 2724: 2722: 2721: 2716: 2711: 2708: 2703: 2700: 2698: 2693: 2691: 2687: 2683: 2679: 2675: 2674:Edmund Gunter 2671: 2667: 2658: 2651: 2647: 2638: 2636: 2631: 2629: 2625: 2624:polycarbonate 2621: 2616: 2614: 2609: 2588: 2583: 2576: 2571: 2564: 2559: 2552: 2547: 2546: 2545: 2543: 2539: 2535: 2525: 2516: 2511: 2504: 2499: 2492: 2487: 2483: 2482: 2475: 2470: 2463: 2458: 2451: 2446: 2439: 2434: 2430: 2426: 2419: 2414: 2413: 2412: 2410: 2409:nautical mile 2406: 2402: 2398: 2394: 2389: 2384: 2383:aspect ratios 2379: 2376: 2372: 2368: 2363: 2359: 2355: 2353: 2348: 2339: 2335: 2333: 2328: 2320: 2310: 2300: 2284: 2280: 2276: 2271: 2267: 2263: 2258: 2254: 2245: 2244:harmonic mean 2241: 2238:(calculating 2223: 2220: 2215: 2210: 2207: 2202: 2197: 2194: 2184: 2175: 2161: 2160: 2158: 2153: 2149: 2145: 2141: 2123: 2114: 2111: 2108: 2105: 2103: 2098: 2094: 2090: 2087: 2082: 2079: 2073: 2065: 2056: 2053: 2050: 2047: 2045: 2040: 2036: 2032: 2029: 2024: 2021: 2015: 2002: 1999:Exploits the 1998: 1997: 1995: 1994: 1993: 1985: 1976: 1974: 1968: 1966: 1962: 1958: 1942: 1939: 1936: 1933: 1923: 1915: 1906: 1892: 1872: 1852: 1848: 1844: 1841: 1838: 1818: 1814: 1810: 1807: 1804: 1801: 1781: 1778: 1775: 1772: 1769: 1766: 1763: 1758: 1754: 1733: 1730: 1727: 1724: 1721: 1718: 1715: 1710: 1706: 1702: 1694: 1685: 1681: 1664: 1660: 1656: 1652: 1643: 1639: 1635: 1631: 1613: 1609: 1599: 1583: 1579: 1564: 1556: 1552: 1551: 1543: 1539: 1536: 1532: 1531: 1528: 1527: 1519: 1499: 1495: 1491: 1488: 1478: 1475: 1472: 1471: 1454: 1450: 1429: 1421: 1418: 1415: 1414: 1396: 1392: 1369: 1366: 1362: 1354: 1351: 1348: 1347: 1343: 1340: 1337: 1336: 1332: 1328: 1325: 1324: 1320: 1316: 1313: 1312: 1308: 1304: 1301: 1300: 1296: 1295:degree–radian 1292: 1288: 1284: 1280: 1277: 1276: 1260: 1252: 1234: 1230: 1225: 1204: 1196: 1193: 1190: 1189: 1185: 1181: 1177: 1174: 1173: 1169: 1165: 1161: 1158: 1157: 1143: 1139: 1136: 1133: 1132: 1128: 1124: 1120: 1117: 1114: 1113: 1109: 1106: 1103: 1102: 1098: 1095: 1092: 1091: 1087: 1084: 1081: 1080: 1077: 1075: 1071: 1067: 1063: 1055: 1051: 1047: 1043: 1042:trigonometric 1039: 1029: 1024: 1014: 1010: 1001: 935: 928: 926: 921: 879: 868: 834: 809: 795: 791: 788:is above the 782: 780: 775: 764: 762: 757: 723: 720: 717: 694: 691: 688: 682: 679: 676: 670: 664: 661: 658: 652: 646: 643: 620: 614: 611: 608: 602: 596: 593: 570: 564: 561: 541: 518: 512: 509: 489: 475: 462: 455: 449: 446: 443: 437: 431: 428: 425: 419: 415: 411: 405: 402: 383: 376: 370: 367: 364: 358: 352: 349: 346: 340: 337: 334: 328: 325: 317: 315: 300: 298: 294: 285: 282: 273: 270: 267: 266: 265: 258: 249: 247: 243: 242:interpolating 238: 236: 232: 228: 224: 220: 216: 212: 207: 204: 200: 196: 192: 189:labeled with 188: 184: 183:ruler's scale 176: 171: 162: 160: 156: 155:United States 151: 149: 145: 141: 137: 133: 129: 125: 121: 117: 113: 108: 106: 102: 98: 94: 90: 84: 82: 78: 74: 70: 66: 62: 58: 54: 51: 47: 43: 39: 30: 26: 22: 5009: 4993: 4926:. Retrieved 4922:the original 4917: 4908: 4899: 4893: 4882:. Retrieved 4878:the original 4873: 4864: 4853:. Retrieved 4849:the original 4844: 4835: 4824:. Retrieved 4819: 4810: 4796: 4784:. Retrieved 4780:the original 4770: 4759:. Retrieved 4755:the original 4751:"Slide Rule" 4745: 4734:. Retrieved 4730:the original 4726:"Slide Rule" 4720: 4709:. Retrieved 4705:the original 4695: 4686: 4677: 4666:. Retrieved 4656: 4645:. Retrieved 4641:the original 4636: 4627: 4616:. Retrieved 4607: 4597: 4588:, May 2006, 4585: 4580: 4568: 4563: 4555: 4547: 4531: 4526: 4518: 4499: 4485: 4477: 4463: 4457: 4449: 4435: 4429: 4421: 4413: 4407: 4396:. Retrieved 4389:the original 4360: 4356: 4343: 4332:. Retrieved 4322: 4305: 4301: 4295: 4286: 4277: 4266:. Retrieved 4257: 4246:. Retrieved 4237: 4204: 4200: 4194: 4183:. Retrieved 4179:the original 4174: 4165: 4154:. Retrieved 4150:the original 4145: 4136: 4125:. Retrieved 4121:the original 4116: 4106: 4098: 4093: 4082:. Retrieved 4078:the original 4068: 4057:. Retrieved 4053:the original 4043: 3997:(5): 80–87. 3994: 3990: 3980: 3958:(12): 8–10. 3955: 3951: 3941: 3930: 3924: 3916: 3912: 3904:the original 3898: 3891: 3883: 3878: 3851: 3848:"Slide Rule" 3841: 3821: 3814: 3803:. Retrieved 3793: 3757: 3746:. Retrieved 3742:the original 3732: 3711: 3687:(1): 14–18. 3684: 3680: 3670: 3661: 3652: 3641:. Retrieved 3637:the original 3633:sphere.bc.ca 3632: 3623: 3612:. Retrieved 3608: 3598: 3587:. Retrieved 3572: 3565: 3548: 3542: 3531:. Retrieved 3526: 3517: 3509: 3503:. Retrieved 3488: 3481: 3473: 3467:. Retrieved 3452: 3445: 3343: 3316: 3292: 3289: 3280: 3260: 3251: 3247: 3243: 3238: 3234: 3208: 3199: 3176: 3147: 3140: 3132: 3118:The typical 3117: 3093: 3085: 3057: 3049: 3041: 3019: 3016: 2921: 2916: 2914: 2907: 2904: 2896: 2881: 2877: 2865: 2863: 2859: 2836: 2829: 2818: 2804: 2801:Vickers Ltd. 2799:airship for 2792: 2787: 2780: 2776: 2772: 2760: 2756: 2749: 2725: 2718: 2712: 2704: 2701: 2694: 2682:Gunter rules 2663: 2632: 2617: 2610: 2607: 2531: 2524: 2479: 2390: 2380: 2364: 2360: 2356: 2349: 2345: 2336: 2329: 2325: 2180: 2148:Gravity Pipe 1991: 1982: 1972: 1969: 1960: 1956: 1924: 1921: 1918:Trigonometry 1912: 1695:of the form 1691: 1682: 1641: 1637: 1633: 1629: 1600: 1570: 1562: 1476: 1420:linear scale 1419: 1352: 1342:linear scale 1341: 1330: 1318: 1306: 1302:Sh, Sh1, Sh2 1294: 1291:small angles 1290: 1286: 1282: 1194: 1183: 1179: 1167: 1163: 1137: 1122: 1118: 1107: 1096: 1085: 1069: 1035: 1017:Other scales 1011: 1007: 936: 929: 922: 885: 869: 859:and then by 832: 810: 793: 789: 783: 776: 765: 758: 481: 318: 311: 291: 277: 263: 239: 227:exponentials 223:square roots 208: 199:lookup table 180: 158: 152: 109: 107:of results. 85: 77:trigonometry 37: 35: 25: 5037:Slide Rules 3410:Slide chart 3302:Collections 3276:3D printing 3248:Deci-Lon 10 2964:actinograph 2884:actinograph 2843:Buzz Aldrin 2832:stethoscope 2821:Lucy Hayner 2805:calculators 2763:engineering 2738:Modern form 2732:declination 2666:John Napier 2597: 1890 1477:Pythagorean 1297:conversion 1066:exponential 823:represents 806:1.4×10 = 14 191:precomputed 187:graduations 175:index cards 132:engineering 120:error-prone 116:John Napier 5074:Logarithms 5053:Categories 4928:2023-01-07 4884:2019-05-01 4855:2019-05-01 4845:MIT Museum 4826:2019-05-01 4786:2012-01-17 4761:2015-04-08 4736:2015-04-08 4711:2010-02-20 4668:2010-02-20 4647:2014-04-21 4618:2023-09-23 4571:May 2006, 4540:hoi polloi 4534:May 2006, 4398:2016-01-02 4334:2024-06-16 4268:2021-05-02 4248:2021-05-02 4185:2016-09-28 4156:2009-10-06 4127:2010-02-20 4084:2013-09-03 4059:2013-09-03 3932:Slide Rule 3805:2010-02-20 3748:2010-02-20 3723:1612.03955 3694:1612.03955 3643:2007-03-14 3614:2021-07-13 3589:2017-03-21 3533:2019-05-01 3527:MIT Museum 3505:2013-03-30 3469:2013-03-30 3437:References 3319:MIT Museum 3312:MIT Museum 3252:Deci-Lon 5 3186:improve it 3154:examples. 2999:See also: 2788:Slide Rule 2672:. In 1620 1253:of radius 1184:cotangents 1142:reciprocal 1044:, usually 636:. Because 554:at offset 252:Components 231:logarithms 73:logarithms 44:-operated 38:slide rule 4590:pp. 80–87 4573:pp. 80–87 4536:pp. 80–87 4377:0922-5773 4019:0036-8733 3972:0031-9228 3327:nomograms 3284:ThinkGeek 3268:celluloid 3190:verifying 2713:In 1821, 2705:In 1815, 2670:logarithm 2613:celluloid 2604:Materials 2538:Otis King 2393:Breitling 2112:− 2088:− 1940:⁡ 1808:− 1764:− 1492:− 1479:scale of 1367:− 1281:used for 1226:π 1175:T, T1, T2 1064:(ln) and 1038:functions 721:× 692:× 683:⁡ 665:⁡ 647:⁡ 615:⁡ 597:⁡ 565:⁡ 513:⁡ 450:⁡ 444:− 432:⁡ 406:⁡ 371:⁡ 353:⁡ 338:× 329:⁡ 303:Operation 159:slipstick 89:nomograms 65:exponents 4960:Archived 4820:MIT News 4612:Archived 4608:Hackaday 4505:Springer 4497:(1998). 4229:20101166 4221:15061314 4035:16708492 4027:26061456 3782:Archived 3767:Archived 3430:Volvelle 3392:Nomogram 3353:See also 3244:Deci-Lon 3029:HP 9100A 2892:exposure 2814:foolscap 2686:priority 2620:acrylics 2540:and the 2429:imperial 2401:airspeed 2001:identity 1973:decitrig 1287:tangents 1180:tangents 1154:) steps 1138:inverted 882:Division 839:. Since 148:obsolete 61:division 50:slidable 5014:. 1920. 4998:. 1905. 3999:Bibcode 3856:Bibcode 3699:Bibcode 3184:Please 2995:Decline 2825:Braille 2641:History 1794:(where 1278:ST, SRT 1168:cosines 1149:⁄ 1050:tangent 1032:roots). 997:⁠ 985:⁠ 970:⁠ 958:⁠ 952:⁠ 940:⁠ 901:⁠ 889:⁠ 736:. With 297:decades 288:Decades 219:product 153:In the 4511: 4470: 4442: 4385:112881 4383: 4375: 4227: 4219: 4033: 4025: 4017: 3970: 3866: 3829: 3580: 3557:319462 3555: 3496: 3460: 3359:Abacus 3333:. The 3329:, and 3037:CORDIC 2791:) was 2628:Teflon 2536:, the 2425:metric 1965:radian 1191:R1, R2 1119:folded 1115:CF, DF 904:. The 833:second 790:second 768:7×2=14 293:Scales 281:cursor 233:, and 217:, the 97:abacus 75:, and 53:rulers 4392:(PDF) 4381:S2CID 4353:(PDF) 4225:S2CID 4023:JSTOR 3718:arXiv 3689:arXiv 3064:Texas 3053:TI-30 3045:HP-35 3011:TI-30 2635:resin 1283:sines 1164:sines 943:5.5×3 865:1,760 813:88×20 750:3×2=6 201:that 181:Each 124:paper 93:ruler 69:roots 40:is a 4509:ISBN 4468:ISBN 4440:ISBN 4373:ISSN 4217:PMID 4031:PMID 4015:ISSN 3968:ISSN 3864:ISBN 3827:ISBN 3578:ISBN 3553:OCLC 3494:ISBN 3458:ISBN 3344:The 3317:The 3009:The 2910:E6-B 2908:The 2797:R100 2730:and 2405:rate 1831:and 1601:For 1384:and 1293:and 1285:and 1182:and 1166:and 1093:A, B 1082:C, D 1056:(log 1048:and 1046:sine 976:8.25 914:2.75 853:1.76 794:must 740:and 203:maps 185:has 142:was 42:hand 4365:doi 4310:doi 4209:doi 4007:doi 3995:294 3960:doi 3321:in 3188:by 3151:E6B 3112:000 3105:000 3102:500 2962:'s 2886:of 2622:or 2375:GPS 2367:E6B 1937:sin 1289:of 988:5.5 961:5.5 910:5.5 892:5.5 857:100 841:8.8 837:100 802:1.4 742:y=3 738:x=2 680:log 662:log 644:log 612:log 594:log 562:log 510:log 447:log 429:log 403:log 368:log 350:log 326:log 5055:: 5008:. 4992:. 4916:. 4872:. 4843:. 4818:. 4685:. 4635:. 4610:. 4606:. 4517:. 4503:. 4493:; 4476:. 4448:. 4416:. 4379:. 4371:. 4361:25 4359:. 4355:. 4306:38 4304:. 4285:. 4223:. 4215:. 4205:14 4203:. 4173:. 4144:. 4115:. 4029:. 4021:. 4013:. 4005:. 3993:. 3989:. 3966:. 3954:. 3950:. 3862:. 3850:. 3697:. 3685:27 3683:. 3679:. 3660:. 3631:. 3607:. 3525:. 3508:. 3472:. 3130:. 3110:45 2919:. 2857:. 2827:. 2594:c. 2403:, 2299:. 2242:, 1473:P 1416:Ln 1326:Th 1314:Ch 1123:pi 1058:10 1052:, 1000:. 972:×3 920:: 867:. 861:10 849:10 845:88 829:10 825:20 808:. 798:10 756:: 237:. 229:, 225:, 161:. 83:. 71:, 67:, 63:, 59:, 36:A 4931:. 4902:. 4887:. 4858:. 4829:. 4804:. 4789:. 4764:. 4739:. 4714:. 4689:. 4671:. 4650:. 4621:. 4401:. 4367:: 4337:. 4316:. 4312:: 4289:. 4271:. 4251:. 4231:. 4211:: 4188:. 4159:. 4087:. 4062:. 4037:. 4009:: 4001:: 3974:. 3962:: 3956:7 3872:. 3858:: 3835:. 3808:. 3751:. 3726:. 3720:: 3705:. 3701:: 3691:: 3664:. 3646:. 3617:. 3592:. 3559:. 3536:. 3215:) 3209:( 3204:) 3200:( 3182:. 3100:1 2427:/ 2352:π 2285:2 2281:z 2277:= 2272:2 2268:y 2264:+ 2259:2 2255:x 2224:z 2221:1 2216:= 2211:y 2208:1 2203:+ 2198:x 2195:1 2154:. 2124:. 2115:y 2109:x 2106:= 2099:y 2095:) 2091:1 2083:y 2080:x 2074:( 2066:, 2057:y 2054:+ 2051:x 2048:= 2041:y 2037:) 2033:1 2030:+ 2025:y 2022:x 2016:( 1961:k 1957:x 1943:x 1934:k 1893:p 1873:q 1853:a 1849:/ 1845:c 1842:= 1839:q 1819:a 1815:/ 1811:b 1805:= 1802:p 1782:0 1779:= 1776:q 1773:+ 1770:x 1767:p 1759:2 1755:x 1734:0 1731:= 1728:c 1725:+ 1722:x 1719:b 1716:+ 1711:2 1707:x 1703:a 1665:2 1661:/ 1657:y 1653:x 1642:y 1638:x 1634:y 1630:x 1614:y 1610:x 1584:2 1580:x 1500:2 1496:x 1489:1 1455:x 1451:e 1430:e 1397:x 1393:e 1370:x 1363:e 1338:L 1261:r 1249:( 1235:2 1231:r 1205:r 1159:S 1151:x 1147:1 1144:( 1127:π 1125:( 1104:K 1070:e 1068:( 994:2 991:/ 980:3 967:2 964:/ 949:2 946:/ 932:1 918:1 906:2 898:2 895:/ 876:1 872:1 821:2 817:2 786:7 772:7 754:3 746:2 724:y 718:x 698:) 695:y 689:x 686:( 677:= 674:) 671:y 668:( 659:+ 656:) 653:x 650:( 624:) 621:y 618:( 609:+ 606:) 603:x 600:( 574:) 571:y 568:( 542:y 522:) 519:x 516:( 490:x 463:. 459:) 456:y 453:( 441:) 438:x 435:( 426:= 423:) 420:y 416:/ 412:x 409:( 384:, 380:) 377:y 374:( 365:+ 362:) 359:x 356:( 347:= 344:) 341:y 335:x 332:( 23:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index