Usuària:Mzamora2/slide rule

EL regle de calcular fabricat per Gilson el 1931 feia sumes i rectes limitat fins a les fraccions.

Arrels i potències

Hi ha escales de decada única (C i D), decada doble (A i B), i decada triple (K). Per calcular $x^{2}$, per exemple, colocar x a l’escala D i llegeix la seva arrel en l’escala A. Invertint aquest procés permet trobar arrels quadrades, i de manera semblant les potències de 3, 1/3, 2/3, i 3/2. S’ha d’anar amb compte si la base, x, es troba en més d’un lloc a l’escala. Per exemple, hi ha dos nous a l’escala A; per tal de trobar l’arrel quadrada de nou; el segon dóna l’arrel quadrada de 90.

Per problemes del tipus $x^{y}$ cal utilitzar les escales LL. Quan tenim diverses escales LL, cal utilitzar la que tingui una x fixada en ella. Per començar, s’ha d’alinear l’1 que es troba més a l’esquerra en l’escala C amb x amb l’escala LL. Llavors, cal trobar y en l’escala C i es baixa fins l’escala LL amb una x en ella. Aquesta escala indicarà la resposta. Si y està "fora de l’escala," cal colocar $x^{y/2}$ i fer l’arrel utilitzant les escales A i B tal i com està descrit a sobre.

Trigonometria

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

For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C. The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with C or, for angles greater than 45 degrees, CI. Common forms such as $k\sin x$ can be read directly from x on the S scale to the result on the D scale, when the C-scale index is set at k. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/radian. Inverse trigonometric functions are found by reversing the process.

Many slide rules have S, T, and ST scales marked with degrees and minutes. So-called decitrig models use decimal fractions of degrees instead.

Logaritmes i exponencials

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.

The Ln scale was invented by an 11th grade student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent x (in the range 0 to 2.3) on the Ln scale and read e^x on the C (or D) scale and e^–x on the CI (or DI) scale. Pickett, Inc. was given exclusive rights to the scale. Later, the inventor created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of its slide rules.

Adició i subtracció

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

The first method to perform addition and subtraction on the C and D (or any comparable scales) requires converting the problem into one of division. For addition, the quotient of the two variables plus one times the divisor equals their sum:

$x+y=\left({\frac {x}{y}}+1\right)y$

For subtraction, the quotient of the two variables minus one times the divisor equals their difference:

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

This method is similar to the addition/subtraction technique used for high-speed electronic circuits with the logarithmic number system in specialized computer applications like the Gravity Pipe (GRAPE) supercomputer and hidden Markov models.

The second method utilizes a sliding linear L scale available on some models. Addition and subtraction are performed by sliding the cursor left (for subtraction) or right (for addition) then returning the slide to 0 to read the result.

Disseny físic

Regles lineals standard

The length of the slide rule is quoted in terms of the nominal length of the scales. Scales on the most common "10-inch" models are actually 25 cm in length, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflowed. Pocket rules are typically 5 inches. Models a couple of meters long were sold to be hung in classrooms for teaching purposes. [1]

Typically the divisions mark a scale to a precision of two significant figures, and the user estimates the third figure. Some high-end slide rules have magnifying 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.

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.

Regles de càlcul circulars

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

A simple circular slide rule, made by Concise Co., Ltd., Tokyo, Japan, with only inverse, square and cubic scales. On the reverse is a handy list of 38 metric/imperial conversion factors.

Breitling Navitimer wristwatch with circular slide rule.

Circular slide rules come in two basic types, one with two cursors (left), and another with a movable disk and a single cursor (right). The dual cursor versions perform multiplication and division by maintaining a fixed angle between the cursors as they are rotated around the dial. The single cursor version operates more like the standard slide rule through the appropriate alignment of the scales.

The basic advantage of a circular slide rule is that the longest dimension of the tool was reduced by a factor of about 3 (i.e. by π). For example, a 10 cm circular would have a maximum precision equal to a 30 cm ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be re-oriented 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 length is shortened to make room for end margins.

Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm off-centre of the pivot can result in a 0.2 mm worst case alignment error. The pivot, however, does prevent scratching of the face and cursors. The highest accuracy scales are placed on the outer rings. Rather than "split" scales, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale.

The main disadvantages of circular slide rules are the difficulty in locating figures along a rotating disc, and limited number of scales. Another drawback of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch.

One slide rule remaining in daily use around the world is the E6B. This is a circular slide rule first created in the 1930s for aircraft pilots to help with dead reckoning. With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The so-called "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions, the E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of its mastery.

In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer–nautical mile and gallon–liter fuel amount conversion functions.

Regles de càlcul cilíndriques

Otis King

There are two main types of cylindrical slide rules: those with helical scales such as the Fuller, the Otis King and the Bygrave slide rule, and those with bars, such as the Thacher and some Loga models. In either case, the advantage is a much longer scale, and hence potentially higher accuracy, than a straight or circular rule.

Materials

Traditionally slide rules were made out of hard wood such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument was made of steel.

In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable, strong and naturally self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933 [2], and probably only a little earlier in Germany. Scales were made of celluloid or plastic. Later slide rules were made of plastic, or aluminium painted with plastic. Later cursors were acrylics or polycarbonates sliding on Teflon bearings.

All premium slide rules had numbers and scales engraved, and then filled with paint or other resin. Painted or imprinted slide rules were viewed as inferior because the markings could wear off. Nevertheless, Pickett, probably America's most successful slide rule company, made all printed scales. Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. The recommended cleaning method for engraved markings is to scrub lightly with steel-wool. For painted slide rules, and the faint of heart, use diluted commercial window-cleaning fluid and a soft cloth.

Història

William Oughtred (1575–1660), inventor of the circular slide rule.

The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593–1656), an English mathematician, in a book entitled L'usage de la reigle de proportion en l'arithmetique & geometrie. The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 he combined two Gunter rules, held together with the hands, to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students, but delayed in publishing them, and like Newton, he became involved in a vitriolic controversy over priority, with his one-time student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.

In 1677, Henry Coggeshall created a two-foot folding rule for timber measure, called the Coggeshall slide rule. His design and uses for the tool gave the slide rule purpose outside of mathematical inquiry.

In 1722, Warner introduced the two- and three-decade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.

In 1815, Peter Mark Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.

Forma moderna

The more modern form was created in 1859 by French artillery lieutenant Amédée Mannheim, "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around that time, as engineering became a recognized professional activity, that slide rules came into wide use in Europe. They did not become common in the United States until 1881, when Edwin Thacher introduced a cylindrical rule there. The duplex rule was invented by William Cox in 1891, and was produced by Keuffel and Esser Co. of New York.

Astronomical work also required fine computations, and in 19th century Germany a steel slide rule about 2 meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.

Engineer using a slide rule. Note mechanical calculator in background.

Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession (in the same way that the stethoscope symbolizes the medical profession). German rocket scientist Wernher von Braun brought two 1930s vintage Nestler slide rules with him when he moved to the U.S. after World War II to work on the American space program. Throughout his life he never used any other pocket calculating devices; slide rules served him perfectly well for making quick estimates of rocket design parameters and other figures. Aluminium Pickett-brand slide rules were carried on five Apollo space missions, including to the moon, according to advertising on Pickett's N600 slide rule boxes [3].

Some engineering students and engineers carried ten-inch slide rules in belt holsters, and even into the mid 1970s this was a common sight on campuses. Students also might keep a ten rule for precision work at home or the office while carrying a five-inch pocket slide rule around with them.

In 2004, education researchers David B. Sher and Dean C. Nataro conceived a new type of slide rule based on prosthaphaeresis, an algorithm for rapidly computing products that predates logarithms. There has been little practical interest in constructing one beyond the initial prototype, however. [4]

Calculadores especialitzades

Hurter and Driffield's actinograph

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

John Rabone & Sons 1892 Cattle Gauge

There were many specialized slide rules for photographic applications; for example, the actinograph of Hurter and Driffield was a two-slide boxwood, brass, and cardboard device for estimating exposure from time of day, time of year, and latitude.

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

In World War II, bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the U.S. Navy actually designed a generic slide rule "chassis" with an aluminium body and plastic cursor into which celluloid cards (printed on both sides) could be placed for special calculations. The process was invented to calculate range, fuel use and altitude for aircraft, and then adapted to many other purposes.

Declinar

TI-30

The importance of the slide rule began to diminish as electronic computers, a new but very scarce resource in the 1950s, became widely available to technical workers during the 1960s. The introduction of Fortran in 1957 made computers practical for solving modest size mathematical problems. IBM introduced a series of more affordable computers, the IBM 650 (1954), IBM 1620 (1959), IBM 1130 (1965) addressed to the science and engineering market. The BASIC programming language (1964) made it easy for students to use computers. The DEC PDP-8 minicomputer was introduced in 1965.

Computers also changed the nature of calculation. With slide rules, there was a great emphasis on working the algebra to get expressions into the most computable form. Users of slide rules would simply approximate or drop small terms to simplify the calculation. Fortran allowed complicated formulas to be typed in from textbooks without the effort of reformulation. Numerical integration was often easier than trying to find closed form solutions for difficult problems. The young engineer asking for computer time to solve a problem that could have been done by a few swipes on the slide rule became a humorous cliché. Many computer centers had a framed slide rule hung on a wall with the note "In case of emergency, break glass."

Another step toward the replacement of slide rules with electronics was the development of electronic calculators for scientific and engineering use. The first included the Wang Laboratories LOCI-2, introduced in 1965, which used logarithms for multiplication and division and the Hewlett-Packard HP-9100, introduced in 1968. The HP-9100 had trigonometric functions (sin, cos, tan) in addition to exponentials and logarithms. It used the CORDIC (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.

The era of the slide rule ended with the launch of pocket-sized scientific calculators, of which the 1972 Hewlett-Packard HP-35 was the first. Such calculators became known as "slide rule" calculators, since they could perform most, or all the functions of a slide rule. Introduced at US$395, even this was considered expensive for most students. But by 1975, basic four-function electronic calculators could be purchased for less than $50. By 1976 the TI-30 offered a scientific calculator for less than $25. After this time, the market for slide rules dwindled quickly as small scientific calculators became affordable.

Avantatges

• 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 . Since you must explicitly note the order of magnitude at each step in order to interpret your results, you are less likely to make wildly wrong errors. You are forced to use common sense and an understanding of your subject as you calculate. Since order of magnitude gets the greatest prominence when using a slide rule, and precision is limited only to the few digits that are normally useful, users are less likely to make errors of false precision.
• When performing a sequence of multiplications or divisions by the same number, the answer can often be determined by merely glancing at the slide rule without any manipulation. This can be especially useful when calculating percentages, e.g., for test scores, or when comparing prices, e.g., in dollars per kilogram. Multiple speed-time-distance calculations can be performed hands-free at a glance with a slide rule.
• A slide rule does not depend on electricity.
• A person can make a slide rule from wood, cardboard, or paper using ordinary tools.
• Slide rules are highly standardized, so there is no need to relearn anything when switching to a different rule.

An advantage of using a slide rule together with an electronic calculator is that an important calculation can be checked by doing it on both; because the two instruments are so different, there is little chance of making the same mistake twice.

Desavantatges

• The typical precision of a slide rule is about three decimal places. A typical pocket calculator displays results to seven or more decimal places.
• A slide rule requires the user to mentally calculate the order of magnitude of the results. For example, 1.5 × 30 (which equals 45) will show the same result as 1,500,000 × 0.03 (which equals 45,000). This 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).
• Errors may arise from mechanical imprecision in slide rules that are warped by heat or use or that were poorly constructed to begin with.

Finding and collecting slide rules

There are still peoplePlantilla:Who who prefer a slide rule over an electronic calculator as a practical computing device. Many others keep their old slide rules out of a sense of nostalgia, or collect slide rules as a hobby.

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

Although there is a large supply of slide rules circulating on the market, specimens in good condition tend to be surprisingly expensive. Many rules found for sale on online auction sites are damaged or have missing parts, and the seller may not know enough to supply the relevant information. Replacement parts are scarce, and therefore expensive, and are generally only available for separate purchase on individual collectors' web sites. The Keuffel and Esser rules from the period up to about 1950 are particularly problematic, because the end-pieces on the cursors, made of celluloid, tend to break down chemically over time.

In many cases, an economical method for obtaining a working slide rule is to buy more than one of the same model, and combine their parts.

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. And in September 2009, on-line retailer ThinkGeek introduced its own brand of straight slide rules, which they describe as "faithful replica[s]" that are "individually hand tooled" due to a stated lack of any existing manufacturers.. The E6B circular slide rule used by pilots has been in continuous production and remains available in a variety of models.

...........................

• "Mad Slide Ruling" — from The Mad Slider Ruler
• International Slide Rule Group — The International Slide Rule Group "ISRG" is devoted to collectors of slide rules and associated mechanical calculating instruments.
• Pastore, Giovanni, Antikythera E I Regoli Calcolatori, Rome, 2006, privately published
• International Slide Rule Museum — Simulator with an Illustrated Self-Guided Course On How To Use The Slide Rule
• Cold War Calculators Slide rules for military and civil defence use.
• "What can you do with a slide rule?" — Reference and use tricks.

How-To's:

• International Slide Rule Museum — Illustrated Self-Guided Course On How To Use The Slide Rule (with simulators)
• "Slide rule construction plans" (PDF) — from Scientific American magazine, May 2006 (accompanying an article by Cliff Stoll)
• "Make your own slide rule" (PDF) — By Luis Fernandes, Dept. of Electrical and Computer Engineering, Ryerson University
• "Make your own circular slide rule" — By Dr. Charles Kankelborg, Department of Physics, Montana State University
• How to make your own slide rule, from Sphere Research
• Photo-realistic Aristo Multilog Nr. 970 Slide Rule Simulator, with digital readout of scales to aid learning, by Stefan Vorkoetter
• A Collection of Photo-realistic Pickett Slide Rule Simulators, by Derek Ross
• Lessons and Videos on using the Slide Rule from The Mad Slide Ruler
• A virtual slide rule for kids Includes lessons such as how to add with slide rule, a simple calculus example using mouse and cheese, and more.