Slide rule, or slide rule, usually refers to logarithmic slide rule. It is an analog computer, which usually consists of three interlocking scale bars and a sliding window (called cursor). It was widely used before 1970, and was later replaced by electronic calculator, becoming an outdated technology.
basic concept
In its most basic form, the ruler uses two logarithmic scales for multiplication and division, which is a time-consuming and error-prone common operation on paper. The user determines the position of the decimal point in the result through estimation. In calculations involving addition, subtraction, multiplication and division, addition and subtraction are carried out on paper, not on a ruler.
In fact, even the most basic student ruler is far more than two scales. Most straightedge consists of three straight bars, which are arranged in parallel and locked with each other, so that the middle bar can slide along the length direction relative to the other two bars. The outer two are fixed, so their relative positions remain unchanged. Some rulers ("double-sided" type) have scales on both sides of the ruler and slider, some have scales on one side of the outer strip and both sides of the slider, and others have scales on only one side ("single-sided" type). The sliding marker has one or more vertical alignment lines, which can be used to record intermediate results on any scale, and can also be used to find corresponding points on non-adjacent scales.
More complex rulers can perform other calculations, such as square roots, exponents, logarithms and trigonometric functions.
Usually, mathematical calculations are made by aligning the marks on the sliding bar with the marks on other fixed bars, and the results are read by observing the relative positions of other marks on the bars.
calculate
increase
The following figure shows a simplified rule with two logarithmic scales. That is to say, a number X is printed where the distance from the "index" of each ruler (marked by the number 1) is proportional to the log X. ..
Logarithm transforms multiplication and division into addition and subtraction thanks to two laws: log(xy) = log(x)+log(y) and log(x/y) = log(x)-log(y). Slide the top scale to the right by the distance of log(x), and align each number y (at the position of the top scale log(y)) with the position of the bottom scale log(x)+log(y). Because log(x)+log(y) = log(xy), this position of the bottom scale is marked as xy, which is the product of x and y.
The figure below shows twice as many as any other number. The index of the upper scale (1) is aligned with 2 of the lower scale. This will shift the entire upper limit to the right by the distance of log(2). The number (multiplier) on the upper scale corresponds to the product on the lower scale. For example, the product of upper scale 3.5 and lower scale 7 is aligned, while 4 and 8 are aligned, and so on, as shown in the figure:
The operation may be "out of range". For example, the above figure shows that 7 on the upper scale is not aligned with any number on the lower scale, so it will not give 2? The answer to 7. In this case, the user can move the upper limit a little to the left and multiply it by 0.2 instead of 2, as shown in the following figure:
Here, the user of the ruler must remember to adjust the decimal point accordingly in order to get the final answer. We need to find two? 7, but we actually calculated 0.2? 7 = 1.4。 So the real answer is 14 instead of 1.4. ...
separate
The figure below shows the calculation of 5.5/2. The top scale 2 is above the bottom scale 5.5. The top 1 is just above quotient 2.75. ..
Other operations
Besides logarithmic scale, some rulers have other mathematical functions on auxiliary scale. The most common ones are trigonometric functions, usually sine and tangent, commonly used logarithm (log 10) (logarithm used to take values on multiplier scale), natural logarithm (ln) and exponential function (ex) scales. Some rulers include Pythagoras ruler for calculating the side length of triangle and ruler for calculating circle. Others have scales for calculating hyperbolic functions. On a ruler, scales and their marks are highly standardized, and the main change lies in which scales are included and the order in which they appear. ：
A, b double logarithmic scale
C, d simple logarithmic scale
K 3 10 logarithmic scale
The scales of CF, DF C and D start from π instead of 1.
CI, DI, DIF reciprocal scale, from right to left
S is used to find sine and cosine on the d scale.
T is used to find the tangents on the d and DI scales.
ST is used for sine and tangent of small angle.
L linear scale, together with c and d scales, is used to find the logarithm of radix 10 and the power of 10.
A set of logarithmic scales used to find natural logarithms and exponents.
A k &；; E 408 1-3 front and back of ruler.
Find the root and strength
There are single tens (C and D), double tens (A and B) and thirty (K) scales. For example, to calculate X 2, we can find X on D and read its square from A, and in turn, we can calculate the square root, the power of 3, 1/3, 2/3, 3/2. You must be careful when looking for the bottom x on the scale, sometimes there will be more than one X. For example, there are two 9s on the A scale. To find the square root of 9, you must use the first 9; Using the second 9 will get the square root of 90.
trigonometric function
For angles between 5.7 and 90 degrees, the sine value can be determined by comparing the S scale with C or D.. The S scale has a second set of angles (sometimes with different colors), which increase in opposite directions and are used to calculate the cosine value. The tangent can be compared with T scale, C scale and D scale, or with CI scale for angles greater than 45. Sine and tangent values of angles less than 5.7 degrees can be obtained by ST scale. The inverse trigonometric function can be obtained by the inverse process.
Logarithmic sum exponent
Logarithm and exponent based on 10 can be found by L scale, which is linear. Use LL scale when the base is e.
structural design
Standard direct rule
The length of the ruler is based on the length of the ruler, not the length of the whole equipment. The most common high-end ruler is 10 inch double ruler, while the student ruler is often 10 inch single ruler. Pocket rulers are usually 5 inches long.
Usually the dividing line is marked to the precision of two significant digits, and then the user estimates the third digit. Some high-end rulers have cursors with magnifying glasses, which can double the accuracy and make 10 inch rulers as easy to use as 20 inch rulers.
There are some techniques that can be used to increase convenience. Triangular ruler sometimes has two marks, one black and one red, indicating the complementary angle, which is the so-called "darmstadt" style. Double rulers often copy some scales on the back. Calibration is usually "split" to achieve higher accuracy.
Special rulers are designed for different engineering, commercial and banking purposes. These usually use special proportions to directly express common calculations, such as loan calculation, optimal purchase quantity or special engineering equations.
Circular ruler
There are two basic types of circular rulers. One has two cursors, and the other has a removable disk and a cursor. The basic advantage of compasses is to reduce the longest dimension to about 3 times (π times). For example, a 10 cm round ruler has the same accuracy as a 30 cm ordinary ruler. Compass also saves the calculation of "out of bounds" because the scale is designed to be "left and right"; When the results are close to 1.0, they never need to be redirected-the ruler is always in bounds.
Round rulers are mechanically stronger, smoother and more accurate than straight rulers, because they only rely on a central bearing. The central support rarely falls off. Bearings also avoid scratching surfaces and cursors. Only the most expensive straightedge provides these functions.
The highest precision scale is placed on the outermost ring. High-end compasses do not use "split" scale, and more difficult scales use spiral scale (such as double logarithmic scale). An eight-inch advanced round ruler can have a double logarithmic scale of 50 inches!
Technically, the real disadvantage of compasses is that the less important scales are closer to the center, so the accuracy is poor. Historically, the main disadvantage of circular ruler is nonstandard. Most students learn how to use a ruler on a ruler, and then they don't think it is necessary to change it to a round ruler.
The rule still in use in the world today is E6B. This is a circular ruler first made in 1930' s, which is used to help airplane pilots calculate dead reckoning algorithm. This is still available in all flight stores and is still widely used. Although GPS has reduced the use of dead reckoning in aviation, E6B is still used as the first choice or dead reckoning instrument, and most flight schools regard its mastery as their learning requirements.
1952, Breitling, a Swiss watch company, introduced a pilot watch with an integrated circular scale for calculating flight time: Breitling Navitimer. Known by Breitling as "aviation computer", Navitimer circle ruler has the functions of flight speed, climbing speed, flight time, distance and fuel consumption, as well as the conversion function of kilometer-nautical mile and gallon-liter fuel capacity.
material
Traditionally, rulers are made of hardwood, such as mahogany or boxwood, plus glass or metal chutes. 1895, a Japanese company began to use bamboo as a ruler. The advantage of bamboo is that it is not sensitive to temperature and humidity. These bamboo rulers were introduced to Sweden in the autumn of 1933 [/p/articles/mi _ qa3950/is _ 200401/ai _ n9372466].
Fermi ultra-long slide rule
In the 1940s, Li Zhengdao studied theoretical physics from Fermi. In order to calculate the temperature of the sun center, Fermi helped Li Zhengdao to make a special ruler 2 meters long.
merits and demerits
This rule tends to correct the errors of "false precision" and significant figures. Usually, the accuracy of ruler users is 3 digits. This is consistent with the data used in most engineering formulas (for example, the material strength is accurate to 2-3 digits, and there are a large number of safety factors-the typical value is greater than 1.5 times-as additional corrections for the errors and changes of building level and the changes of materials). When using a modern pocket calculator, the precision is displayed as 7 to 10, but in fact, the result cannot be more accurate than the input number.
Rulers need to estimate the order of magnitude of the results at all times. On the ruler, 1.5? 30 (45) and 1, 500,000? 0.03 (equal to 45,000) gives the same result. It depends on the engineer's continuous evaluation of the "validity" of the results: this often does not exist in the use of computer programs or calculators. For example, an employee who cannot judge the rationality of numbers may be operating a calculator.
When calculating a series of multiplication or division, and the factors are the same, you can directly scan the answer from the ruler without any operation. For example, on the ruler in the picture above, you can calculate any multiplication by 2, just look at it and don't use your hands. This is useful when calculating percentages, such as exam scores.
This ruler does not need batteries.
Rulers are different from electronic calculators. They are highly standardized. You don't need to learn anything if you change another one.
The advantages of using a ruler besides an electronic calculator are: an important calculation can be checked after two calculations; Because the two instruments are too different, it is unlikely to make the same mistake twice.
Disadvantages: The biggest disadvantage of the slide rule is that it cannot be added or subtracted, and it must be added or subtracted with an abacus or other auxiliary tools.
The slide rule is in China.
Emperor Kangxi was the first person to use the slide rule in the history of China. He made a Gantt slide rule out of ivory.
Before the 1970s, all science and engineering students in China had a hand, which was an essential computing tool. The "natural double slide rule" produced by Shanghai slide rule factory is a reference to Cofell &; Ethiopian, another short slide rule, is imitated by German Faber-castel, which is accurate and beautiful to manufacture.
Ke slide rule has no centimeter and millimeter scales; The advantage of German Faber-castel slide rule is that it has two scales, centimeter and millimeter, which can be used for calculation and drawing.
Find and collect rulers
For the reasons given above, some people still prefer to use slide rule instead of electronic calculator as a practical calculation tool. Many other people keep their old rulers out of nostalgia or collect them as hobbies or special decorations.
The most popular model is Keuffel &；; Ethiopia's Deci-Lon is an advanced scientific and engineering slide rule, which can be divided into 10 "ordinary" type (Deci-Lon 10) and 5 "pocket" type (Deci-Lon 5). Another popular American model is the 8-inch scientific instrument round ruler. Among European models, Faber-castel's high-end models are the most popular among collectors.
Although there are a large number of rulers in circulation in the market, well-preserved specimens are often surprisingly expensive. Many rulers sold on online auction websites are broken or missing parts. Replacement parts are rare, so they are expensive, and are usually only sold sporadically on the websites of individual collectors. The model before Cofell & Esser1950 is particularly problematic, because the cursor end will be damaged by chemical reactions over time. In many cases, the most economical way to get a usable ruler is to buy more rulers of the same model and then assemble their parts.
The best place to find the slide rule is the flea market. You can usually buy a well-preserved Ke or Faber-castel slide rule for 2 dollars.
annotations
.. resetting scales is not handled like 2? 7. The only way to multiply this out of range; Other methods are: (1) using double decimal system. (2) Use a folding scale. In this example, it is sufficient to align 1 of the C scale with 2 of the D scale. Move the cursor to 7 of CF, and then read the result from DF. (3) Use CI scale. Put 7 on CI above 2 on D scale, and then read the result from 1 on D scale aligned with CI scale. Because 1 appears twice on CI, there is always one in the range. The method 1 is easy to understand, but it will bring the loss of accuracy. The advantage of method 3 is that it only uses two rulers.
.. there are several ways to divide. The advantage of the method given here is that the final result will not cross the line, because one of the two ends 1 can be selected.