The slide rule , also known colloquially in the United States as slipstick , is a mechanical analog computer. Slide rules are used primarily for multiplication and division, as well as for functions such as exponents, roots, logarithms and trigonometry, but usually not for addition or subtraction. Although similar in name and appearance with standard ruler, slide rules are not intended to be used to measure long or draw straight lines.
Slide rules exist in a variety of styles and generally appear in linear or circular form with a series of standard (scale) signals essential for performing mathematical calculations. Slide rules created for specific fields such as aviation or finance typically feature additional scales that assist in the general calculations for those fields.
At the simplest, each multiplied number is represented by a length on a sliding ruler. Since each ruler has a logarithmic scale, it is possible to align it to read the number of logarithms, and therefore calculate the product, of two numbers.
Pastor William Oughtred and others developed the slide rule in the 17th century based on works that appeared on logarithms by John Napier. Before the advent of electronic calculators, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow during the 1950s and 1960s even as computers were being introduced gradually; but circa 1974 handheld electronic scientific calculators make the most obsolete and most suppliers leave the business.
Video Slide rule
Basic concepts
In its most basic form, slide rules use two logarithmic scales to allow for quick multiplication and division of numbers. This general operation can be time-consuming and error-prone when done on paper. More complicated slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.
Scale can be grouped over several decades, ranging from 1 to 10 (eg 10 n to 10 n 1 ). Thus the single decade scale of C and D ranges from 1 to 10 across the width of the slide rule while the double decade scales A and B range from 1 to 100 over the width of the slide rule.
In general, mathematical calculations are performed by aligning the marks on the center strip with a mark on one fixed strip, and then observing the relative positions of other marks on the strip. A number that is aligned with a mark assigning an approximate value to a product, a share, or other calculation result.
The user determines the location of the decimal point in the results, based on mental estimation. Scientific notation is used to track decimal points in more formal calculations. The steps of addition and subtraction in calculations are generally done mentally or on paper, not on slide rules.
Most slide rules consist of three linear strips of the same length, parallel in parallel and interlocked so that the center strip can be moved elongated relative to the other two. The two outermost strips are fixed so that their relative positions remain unchanged.
Some slide rules ("duplex" models) have scales on both sides of the rule and the slide strip, others on one side of the outer strip and both sides of the slide strip (which can usually be pulled out, reversed and re-inserted for convenience), others on one side only (the "simplex" rule). Slide cursors with vertical alignment lines are used to find the appropriate point on a scale that is not adjacent to each other or, in the duplex model, is on the other side of the rule. The cursor can also record intermediate results on any scale.
Maps Slide rule
Operation
Multiplication
Operations can be "not scaled"; for example, the diagram above shows that the slide rule does not position 7 on the upper scale above any number on a lower scale, so it does not give an answer for 2ÃÆ' â € "7. In such cases, the user can shift the top to left the right index is parallel to 2, effectively dividing by 10 (by reducing the full length of the C-scale) and then multiplying by 7, as in the illustration below:
Here the slide rule users should remember to adjust the decimal point appropriately to correct the final answer. We want to find 2ÃÆ' â € "7, but we count (2/10) ÃÆ' â €" 7 = 0.2ÃÆ' â € "7 = 1,4. So the correct answer is not 1.4 but 14. Resetting the slide is not the only way to handle multiplication that will produce insignificant results, such as 2ÃÆ' â € "7; some other methods are:
- Use the double decade scales A and B.
- Use the folded scales. In this example, set left 1C opposite to 2 from D. Move cursor to 7 on CF, and read the result from DF.
- Use CI upside-down scale. Position 7 on the CI scale above 2 on the D scale, and then read the results of the D scale below 1 on the CI scale. Since 1 occurs in two places on a CI scale, one will always be on a scale.
- Use CI reversed scale and scale C. Align 2 CI with 1 D, and read the result from D, below 7 on C scale.
- Uses circular slide rules.
Method 1 is easy to understand, but it contains loss of precision. Method 3 has advantages that involve only two scales.
Division
The illustration below shows the calculations of 5.5/2. 2 on the upper scale placed above 5.5 on the lower scale. 1 on the top scale lies above the quotient, 2.75. There is more than one method for sharing, but the method presented here has the advantage that the end result can not be out of scale, because one has the option of using 1 at both ends.
More operations
In addition to the logarithmic scale, some slide rules have other math functions encoded on other additional scales. The most popular are trigonometric, usually sine and tangent, common logarithm (log 10 ) (for taking log values ​​on multiplier scales), natural logon (ln) and exponential ( e x ) scale. Some rules include Pythagoras scales, to find the sides of a triangle, and a scale to describe the circle. Others have scales to calculate hyperbolic functions. In linear rules, their scale and labeling are highly standardized, with variations usually only occurring in the included scale and in what order:
The Binary Slide Rule produced by Gilson in 1931 undertook a limited addition and subtraction function to the fraction.
Root and power
There are single decades (C and D), double decades (A and B), and three decades (K) scale. To calculate
For problem, use the LL scale. When some LL scales are present, use one with x on it. First, align the leftmost 1 on a scale C with x on the LL scale. Then, find y on a scale C and down to the LL scale with x above it. The scale will show the answer. If y "is out of scale," find and square using the A and B scales as described above. Or, use the rightmost 1 on the C scale, and read the answer from the next higher LL scale. For example, align the most right 1 on a scale C with 2 on a scale of LL2, 3 on a scale line C to 8 on the LL3 scale.
Trigonometry
The S, T, and ST scales are used for trigonometric functions and multiples of trigonometric functions, for angles in degrees.
For an angle of about 5.7 to 90 degrees, the sinus is found by comparing the scale of S to scale C (or D); though in many closed-body rules, the S scale relates to scale A instead, and the following should be adjusted appropriately. Scale S has a second set of angles (sometimes in different colors), which run in the opposite direction, and used for the cosine. Tangents were found by comparing the T scale with a scale of C (or D) for angles less than 45 degrees. For angle greater than 45 degrees, the CI scale is used. Common shapes like can be read directly from x on the S scale for the result at scale D, when the C-scale index is set to k . For angles below 5.7 degrees, the sinuses, tangent lines, and radians are approximately the same, and are found on the scale of ST or SRT (sinus, radians, and tangent lines), or only divided by 57.3 degrees/radians. The reverse trigonometry function is found by reversing the process.
Many slide rules have S, T, and ST scales characterized by degrees and minutes (eg some Keuffel and Esser models, Teledyne-Post Mannheim-type final models). The so-called decitrig model uses decimal fractions of degree instead.
Loganithms and exponential
The base-10 and exponential logarithms are found using L-scale, which is linear. Some slide rules have an Ln scale, which is for base e. Logarithms to other bases can be calculated by reversing the procedure to calculate the strength of a number. For example, the log2 values ​​can be determined by marching the far left or rightmost 1 on a C scale with 2 on the LL2 scale, finding the logarithmic number calculated on the corresponding LL scale, and reading the log2 value on the C scale.
Addition and subtraction
Slide rules are usually not used for addition and subtraction, but still may be done using two different techniques.
The first method for adding and subtracting C and D (or comparable scaling) requires changing the problem into one of the divisions. In addition, the outcome of the two variables plus one divider is equal to their number:
Untuk pengurangan, hasil bagi dua variabel dikurangi satu kali pembagi sama dengan perbedaannya:
This method is similar to the addition/subtraction techniques used for high-speed electronic circuits with logarithmic number systems in specialized computer applications such as Gravity Pipe Supercomputers (GRAPE) and hidden Markov models.
The second method using linear scaling of L shear is available on some models. Addition and subtraction is done by moving the cursor to the left (for subtraction) or right (for addition) then returning the slide to 0 to read the result.
Physical design
Standard linear rules
The slide rule width is quoted in terms of the nominal width of the scale. The scale of the most common "10-inch" model is actually 25 cm, as it is made for standard metrics, although some rules offer slightly extended scales to simplify manipulation when results overflow. The pocket rule is usually 5 inches. Models as wide as several meters are sold for hanging in the classroom for teaching purposes.
Usually division marks scale to the precision of two important numbers, and the user estimates the third number. Some high-end slide rules have a magnifying cursor that makes the marks easier to see. The cursor can effectively double readout accuracy, allowing 10-inch slide rules to serve as well as 20-inches.
Various other conveniences have been developed. Trigonometric scales are sometimes labeled double, black and red, with complementary angles, called "Darmstadt" styles. Duplex slide rules often duplicate some scales on the back. Scales are often "separate" to get higher accuracy.
Circular slide rule
Circular slide rules come in two basic types, one with two cursors (left), and another with a free plate and one cursor (right). Double cursor versions do multiplication and division by holding fast corners between cursors as they rotate around buttons. A one-time cursor version operates more like a standard slide rule through proper scale alignment.
The basic advantage of circular slide rules is that the widest dimensions of the tool are reduced by a factor of about 3 (ie by?). For example, a 10 cm circle will have approximately a maximum precision equal to the usual slide rule of 31.4 cm. The circular slide rule also eliminates "off-scale" calculations, since the scale is designed to "wrap"; they should not be reoriented when the results are close to 1.0 - rules are always on a scale. However, for non-cyclic non-spiral scales such as S, T, and LL's, the width of the scale is narrowed to make room for the final margin.
Round-circular rules are mechanically rougher and smoother-moving, but their scale-alignment precision is sensitive to the center of the central axle; one minute 0.1 mm off-center of the pivot can produce the worst case alignment error of 0.2 mm. However, the pivot prevents facial scratches and cursors. The highest accuracy scale is placed on the outer ring. Instead of a "divided" scale, upper class circular rules use spiral scales for more complex operations such as log-of-log scales. An eight-inch premium circular rule has a 50-inch spiral log-log scale.
The main disadvantage of circular slide rules is the difficulty in finding numbers along the course of the meal, and the number of scales is limited. Another disadvantage of the circular slide rule is that the less important scale is closer to the center, and has lesser precision. Most students learn to use slide rules on linear slide rules, and find no reason to switch.
One slide rule left in everyday use worldwide is E6B. This is a circular slide rule first made in the 1930s for aircraft pilots to help with dead calculations. With the help of the scales printed on the frame it also helps with other tasks such as changing the time, distance, speed, and temperature value, compass errors, and calculating fuel usage. The so-called "prayer wheels" are still available in airline stores, and are still widely used. While GPS has reduced the use of dead reckoning for air navigation, and handheld calculators have taken over many of its functions, the E6B is still widely used as a primary or backup device and the majority of aviation schools demand that their students have a certain level of proficiency. in its use.
The Proportion Wheel is a simple circular slide rule used in graphic design to calculate aspect ratios. Setting the original values ​​and desired sizes on the inner and outer wheels will display their ratio as a percentage in a small window. They are not as common as they appear since computerized layout, but is still created and used.
In 1952, the Swiss watch company Breitling introduced a pilot watch with an integrated circular slide rule specifically for flight calculation: Breitling Navitimer. The Navitimer's circular rules, referred to by Breitling as "navigation computers," show air velocity, climbing/ascent rate, flight time, distance, and fuel consumption function, as well as kilometer - nautical miles and gallons -.
Cylindrical slide rules
There are two main types of slide cylinder rules: those with helical scales like Fuller, Otis King and Bygrave slide rules, and those with bars, such as Thacher and some Loga models. In either case, the advantage is a much longer scale, and therefore potentially more precise, than that provided by straight or circular rules.
Materials
Traditional slide rules are made from hardwoods such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument is made of steel.
In 1895, a Japanese company, Hemmi, began to create a slide rule of bamboo, which has a dimensionally stable, strong, and natural self-lubricating advantage. This bamboo slide rule was introduced in Sweden in September, 1933, and probably only slightly earlier in Germany. Scales made of celluloid, plastic, or painted aluminum. Then the cursor is acrylic or polycarbonate glide on the Teflon bearings.
All premium slide rules have numbers and scale engraved, and then filled with paint or other resins. Painted or printed slide rules are viewed as inferior because the marks can fade. Nevertheless, Pickett, perhaps the most successful slide rule company in America, makes all print scores. Premium slide rules include smart catches so the rules will not fall apart by chance, and bumpers to protect scales and cursors from scouring on the table surface. The recommended cleaning method for engraved signs is a lightweight scrub with steel wool. For the painted slide rules, use commercial diluted window cleaning fluid and soft cloth.
History
The slide rule was found around 1620-1630, shortly after John Napier's publication of the concept of logarithms. In 1620 Edmund Gunter of Oxford developed a counting device on a single logarithmic scale; with additional gauges that can be used to breed and divide. At c. 1622, William Oughtred of Cambridge incorporates two Gunter hand-held rules to create a device that can be recognized as a modern slide rule. Like his contemporary in Cambridge, Isaac Newton, Oughtred teaches his ideas personally to his students. Also like Newton, he became involved in a fierce controversy over priorities, with his only student Richard Delamain and his previous claims from Wingate. Oughtred's ideas were only published in William Forster's student publications in 1632 and 1653.
In 1677, Henry Coggeshall set up a two-foot fold rule for wood size, called the Coggeshall slide rule, extending the use of slide rules beyond math questions.
In 1722, Warner introduced a scale of two and three decades, and in 1755 Everard included an inverse scale; slide rules containing all these scales are usually known as "polyphase" rules.
In 1815, Peter Mark Roget invented the log slide log rules, which included a scale displaying the logarithm of logarithms. This allows users to directly perform calculations involving roots and exponents. This is very useful for fractional powers.
In 1821, Nathaniel Bowditch, described in the American Practical Navigator "shear rules" which contained the function of trigonometric scales on fixed sections and log-sinus lines and log-tans on the sliders used to solve navigation problems.
In 1845, Paul Cameron of Glasgow introduced a maritime slide rule capable of answering navigational questions, including rising rights and the declination of the sun and the major stars.
Modern shape
A more modern slide rule was created in 1859 by the French artillery lieutenant Amà © dà © Å © e Mannheim, "lucky to have his government made by a national reputation company and in that case adopted by French Artillery." It was at this point that engineering became a recognized profession, resulting in widespread slide rules in Europe - but not in the United States. There, Edwin Thacher's cylinder rules were held after 1881. The duplex rule was invented by William Cox in 1891, and produced by Keuffel and Esser Co. from New York.
The work of astronomy also requires precise calculations, and, in the 19th century in Germany, a steel slide rule of about two meters was used at one observatory. It has a microscope attached, giving it an accuracy of up to six decimals.
Throughout the 1950s and 1960s, slide rules were a symbol of the profession of engineers in the same way stethoscopes were the medical profession.
German rocket scientist Wernher von Braun bought two rules of slides of Nestler in the 1930s. Ten years later he took them with him when he moved to the United States after World War II to work on American space efforts. Throughout his life he never used any other slide rules. He used his two Nestlers on the way to the NASA program that landed a man in July 1969.
The Pickett-brand Aluminum slide rule is done on the Apollo Project space mission. Buzz Aldrin's N600-ES model that flew with it to the moon on Apollo 11 was sold at auction in 2007. The N600-ES model taken on Apollo 13 1970 was owned by the National Air and Space Museum.
Some engineering students and engineers brought ten-inch slide rules in holster belts, a common sight on campuses even up to the mid-1970s. Until the emergence of a pocket digital calculator, students may also retain the ten or twenty-inch rule for precision work at home or office while carrying a five-inch slide rule around them.
In 2004, educational researchers David B. Sher and Dean C. Nataro developed a new type of slide rule based on prosthaphaeresis, an algorithm for fast computing products that precedes logarithms. However, there is little practical interest in building one beyond the initial prototype.
Special calculator
Slide rules are often specific to varying degrees for their usage areas, such as excise duty, evidence counting, engineering, navigation, etc., but some slide rules are very specific for very narrow applications. For example, John Rabone & amp; Sons 1892 lists the catalog "Measuring Tapes and Animal Measurers", a tool for estimating the weight of a cow of its size.
There are many slide rules specific to photography applications; for example, the Hurter and Driffield actinographs are boxwood, brass, and two-slide cardboard boxes to estimate exposures of time, day, and latitude.
Special slide rules were created for various forms of engineering, business, and banking. These often have general calculations that are directly expressed as special scales, such as loan calculations, optimal purchase amounts, or certain engineering equations. For example, the Fisherman Control company distributes custom slide rules tailored to solve the equations used to select the appropriate industrial flow control valve size.
Pilot balloon slide rules are used by meteorologists in weather services to determine the top wind speed of a hydrogen or helium air balloon.
In World War II, bombers and navigators who needed quick calculations often used special slide rules. One US Navy office is actually designing a generic "chassis" slide rule with aluminum bodies and plastic cursors where celluloid cards (printed on both sides) can be placed for special calculations. This process was created to calculate range, fuel use and altitude for aircraft, and then adjusted for many other purposes.
E6-B is a circular slide rule used by pilots and navigators.
The circular slide rule to estimate the date of ovulation and fertility is known as the wheel calculator .
Decline
The importance of the slide rule began to diminish as electronic computers, a new but rare resource in the 1950s, became more widely available to technical workers during the 1960s. (See History of computer hardware (1960s-now).)
The computer also changes the calculation properties. With slide rules, great emphasis is placed on algebra to get the expression into the easiest form to calculate. Users will only estimate or drop a small term to simplify the calculation. FORTRAN allows complicated formulas to be implemented without such reformulation. Numerical integration is often easier than trying to find a closed-form solution for a difficult problem. Young engineers who ask for computer time to solve problems that can be done by some friction on the slide rule into a funny cliche.
One more step of the slide rule is the introduction of a relatively inexpensive electronic desktop scientific calculator. The first included Wang Laboratories LOCI-2, introduced in 1965, using logarithms for multiplication and division; and HP 9100A Hewlett-Packard, introduced in 1968. They are both programmable and supplied exponential and logarithm functions; HP has trig functions (trigonometry (sine, cosine, and tangent) and hyperbolic trigonometric functions as well. HP uses the CORDIC algorithm (digital computer rotation coordinates), which allows to calculate trigonometric functions using only shifts and adding operations. This method facilitates the development of smaller scientific calculators.
As with mainframe computing, the availability of these machines did not significantly influence the use of slide rules everywhere until the cheap available electronic hand calculators available in the mid-1970s, at that time, rapidly declined. Hewlett-Packard HP-35 pocket-sized scientific calculator was the first handheld device of its kind, but it cost US $ 395 in 1972. This can be justified for some engineering professionals but too expensive for most students. In 1975, an electronic calculator of four basic functions could be purchased for less than $ 50, and in 1976, the TI-30 scientific calculator sold for less than $ 25.
Compared to electronic digital calculators
Most people find slide rules that are hard to learn and use. Even during their heyday, they have never been in contact with the general public. The addition and reduction of unsupported operations on slide rules and performing calculations on slide rules tend to be slower than on calculators. This causes engineers to use mathematical equations that favor the easy operation of the slide rule for more accurate but complex functions, this estimate can lead to inaccuracies and errors. On the other hand, the layout, manual operation of slide rules cultivates in the user intuition for numerical and scale relationships that people who have used only digital calculators often lack. The slide rule will also show all the calculation conditions along with the results, eliminating the uncertainty about what the actual calculations are doing.
The slide rule requires the user to separately calculate the order of magnitude of the answer to place the decimal point in the result. For example, 1.5 ÃÆ'â € "30 (which equals 45) will show the same result as 1,500,000 ÃÆ'â €" 0.03 (which equals 45,000). This separate calculation tends not to lead to extreme calculation errors, but forces the user to track the amount in short-term memory (which is error-prone), keep records (complicated) or reason about it at each step (which diverts from other calculation requirements).
The typical arithmetic precision of the slide rule is about three significant digits, compared to many digits on the digital calculator. Because the order of magnitude gets the greatest advantage when using slide rules, users are less likely to make false precision errors.
When doing a sequence of multiplication or division with the same number, the answer can often be determined simply by looking at the slide rule without any manipulation. This can be very useful when calculating percentages (eg for test scores) or when comparing prices (eg in dollars per kilogram). Some speed-time-distance calculations can be done hands-free at a glance with slide rules. Other useful linear conversions such as pounds to kilograms can be easily tagged on the rules and used directly in the calculations.
Being fully mechanical, the slide rules are not dependent on the power grid or the battery. However, mechanical inaccuracies in slide rules poorly constructed or bent by heat or usage will cause errors.
Many seafarers maintain slide rules as a backup for navigation in the event of a power failure or decline in battery on long route segments. Slide rules are still commonly used in flight, especially for smaller aircraft. They are simply being replaced by integrated aviation computers, special and expensive destinations, and not general-purpose calculators. E6B circular slide rules used by pilots have been produced continuously and remain available in various models. Some watches designed for flight use still display the slide rule scale to allow for quick calculations. Citizen Skyhawk AT is a noteworthy example.
Today's slide rule
Even today, some people prefer slide rules rather than electronic calculators as practical computing devices. Others keep their old slide rules from a sense of nostalgia, or collect them as a hobby.
Popular collection models are Keuffel & amp; Esser Deci-Lon , the scientific slide rules and premium techniques are available in either ten inches (25 cm) "ordinary" (Deci-Lon 10 ) and five inch "pockets" ( Deci-Lon 5 ) variant. Another valuable American model is the eight-inch (20 cm) circular scientific instrument. European rules, high-end Faber-Castell models are the most popular among collectors.
Although many slide rules circulate on the market, specimens are in good condition tends to be expensive. Many of the rules found to be sold on online auction sites are damaged or there are missing sections, and the seller may not know enough to provide relevant information. Spare parts are rare, expensive, and generally available only for separate purchases on individual collector websites. The Keuffel and Esser rules from the period up to about 1950 are very problematic, since the final pieces on the cursor, made of celluloid, tend to break down chemically over time.
There are still some sources for the new slide rule. The Concise Company of Tokyo, which started as a circular slide rule producer in July 1954, continues to make and sell it today. In September 2009, ThinkGeek's on-line retailer introduced its own brand of straight slide rules, described as "replica loyalty" that were "individually equipped". This is no longer available in 2012. In addition, Faber-Castell has a number of slide rules that are still in stock, available for international purchase through their web store. The proportional wheel is still used in graphic design.
Various slide rule simulator applications are available for Android and iOS-based smartphones and tablets.
Special slide rules such as the E6B used in flight, and the cannon rules used in artillery deployment are still used though no longer on a regular basis. These rules are used as part of the teaching process and instruction as in learning to use them students also learn about the principles behind the calculations, it also allows students to be able to use this instrument as a backup in case of modern electronics in general use fails.
See also
Note
External links
- General information, history
- International Slide Rule Museum
- History, theory, and use of engineering slide rules - By Dr. James B. Calvert, University of Denver
- Slide Slide House Slide House Page
- Front of Home Page Slide Oughtred Society - Preserved for preservation and slide rule history
- Rod Lovett's Rules of Slide - Comprehensive Aristo Site with lots of search facilities
- Derek virtual slide rule gallery - Javascript historical slide rule simulation
- Ã, "Slide rules". New International Encyclopedia . 1905.
- Ã, "Slide-rule". Encyclopedia Americana . 1920.
- Reglas de CÃÆ'¡lculo - Faber Castell's enormous collection
- Sets of slide rules - French Slide Rules (Graphoplex, Tavernier-Gravet, etc.)
- Eric Slide Rule Site - History and usage
- Slide Rules - Information from the HP Calculator Museum
Source of the article : Wikipedia
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