A slide rule is a simple looking mechanical device based on logarithms that performs a wide range of mathematical operations. Visually, it just looks like an assembly of sliding and stationary scales, but hidden in the design of the scales is extraordinary mathematical sophistication, able to perform simple operations from multiplication or division to complicated fractional powers and roots, and complex trigonometric functions. The range of its power is quite incredible, and few people can even perform its more complex functions manually. For example, how are your manual cube root skills today?
It can work with numbers of any size but can typically provide only 3 significant figures of resolution. It also does not do regular addition and subtraction (get out your abacus for that), although its fundamental design is based on the addition and subtraction of logarithmically related distances.
Precision of 3 significant figures is enough for most engineering design in the real world, as it provides accuracy to 0.1%, quite adequate for serious work, especially in a world where many measurements and parts are rarely even accurate to 1%. Your calculator may produce 8 significant figures, but when the total uncertainty of your data is considered, often only 3 are used.
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The slide rule also has one truly magical property it is a parallel calculator, rather than a serial one like your desktop electronic calculator. Once a relationship is set up on a slide rule, all variations are also set up at the same time, allowing you to have a visualization of changing results with different values, without any additional real work. On your electronic calculator, you will be pressing buttons for hours to see what a slide rule does just by existing. In the engineering world, this is called a Monte Carlo analysis, where the values are varied to see the results, usually with a powerful computer to provide the calculating horsepower. Even the lowliest student slide rule does this automatically.
For a simple example, lets say you are on vacation in India and want to buy an antique statue. Set up the exchange rate of rupees to dollars or your currency of choice on your slide rule, and you can see what it costs, BUT, you also see every other price at that rate at the same time (thus making dickering over the price much easier). This is quite amazing, and impossible for your electronic calculator, but just everyday life for a slide rule.
The most commonly seen slide rules are made of wood or bamboo covered with celluloid or are made entirely of plastic. One leading American company made them of aluminum. Some were made of plain wood or steel or other metals.
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Virtually all 20th Century slide rules have a glass or plastic cursor, or indicator, with a hairline, which slides along its length. The sliding scales allow relationships to be set up, and with a cursor, complex relationships may be used across the entire slide rule body, not just between adjacent scales.
Most slide rules are linear, like a ruler, but some are circular discs.
Some scarce types are cylindrical.
Linear slide rules were made in several scale lengths.
Ten inch scale length is by far the most common. (Actually the length is usually 25 centimeters, but it is normally called a ten inch slide rule.)
Five inch scale length (actually 12.5 centimeters) pocket slide rules are also quite common.
Many major slide rule manufacturers also produced a few examples of their most popular 10 inch rules in a 20 inch (50 cm) size. The additional space between markings on the scales contributes to greater precision and the 20 length certainly creates an imposing desktop presence.
The actual overall length of a slide rule is slightly greater than the scale length, so the cursor does not fall off while working at the scale ends, and to provide space for the end braces on duplex slide rules.
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Slide rules calculate by adding and subtracting logarithmic distances by positioning the slide and body. As a result, they can multiply, divide, do roots and powers, calculate logarithms and a wide variety of trigonometry functions. The mathematical relationships are locked into the scale distances specific to the calculation.
Scales vary greatly. For example, a scale used to calculate tangents (often called a T scale) has very different arrangements of scale numbers and markings than scales used to calculate squares and square roots (often called A and B scales).
Slide rules cannot add or subtract in the conventional way ... though they work by adding and subtracting logarithms in order to multiply and divide. The designers of most slide rules felt that addition and subtraction could be done adequately by manual computation, but this could have been done (with limited range) simply by putting two linear scales on a slide rule.
Scales have been designed and refined over many years to facilitate chained calculation, thus they are very good for quickly solving complex formulas, especially those with many multiplied or divided terms. Slide rules also visually reveal ranges of answers in a way that calculators cannot do.
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A slide rule multiplies by adding logarithms of numbers. It divides by subtracting their logarithms.
Now ... dont go running out the door! Logarithms are simple. Although one does not need to know logarithms to use a slide rule, a basic understanding of them is helpful.
Logarithms are exponents. Exponents are powers to which one may raise a number ... like 42 = 16 (4 to the 2nd power equals 16) and 43 = 64 (4 to the 3rd power = 64). The little 2 and 3 are exponents.
When you multiply numbers, you add their exponents. When you divide, you subtract exponents.
Thus 22 = 4. 23 = 8. 22 x 23 = 25 = 32. And 25 / 22 = 23. 32 / 4 = 8
A slide rule does this by sliding logarithmic scales of numbers alongside each other and thus adding or subtracting the logarithmic distances.
It is very easy to add numbers with a ruler, as shown in the illustration below.
The ruler on top will add 3 to any number on the bottom ruler. For example, look at 2 on the top ruler and read the answer 5 on the bottom ruler. Look at any number on the top ruler and the number on the bottom ruler will be 3 greater.
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Your slide rule adds logarithms in the same way.
At the top of page 8, we see a slide rule adding the logarithm of 2 to the logarithm of 3 to get the logarithm of 6 ... thus multiplying 2 by 3 to get 6. (Remember that when we add exponents, or logarithms, we are multiplying the numbers they represent.)
The common logarithm of a number is the power to which one must raise the number 10 (the base) to obtain the number in question. This power, or exponent, is thus called the base 10 logarithm, or the common logarithm, of the number. Any number can be expressed as ten to some power. (Other bases are sometimes used, but 10 is the most common.)
Using 10 as the base (common logarithms), note the following:
The logarithm of 1 is zero. Log 1 = 0. 100 = 1. 10 to the zero power = 1.
Logarithm of 10 is 1. Log 10 = 1. 101 = 10. 10 to the first power = 10.
All numbers between 1 and 10 may be expressed as 10 to some power between 0 and 1, for they are all between 100 and 101. For example, Log 2 = 0.301. 10 to the 0.301 power = 2. 100.301 =2.
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The logarithm of l00 is 2. Log 100 = 2. 102 = 100. 10 squared = 100. All numbers between 10 and 100 may be expressed as 10 to some power between 1 and 2, for all are between 101 and 102.
The logarithm of 1000 is 3. Log 1000 = 3. 103 = 1000. 10 cubed = 1000.
The logarithm of 0.l is minus 1. Log 0.1 = -1. 10-1 = 1/101 = 1/10 = 0.1
Logarithm of 0.01 is minus 2. Log 0.01 = -2. 10-2 = 1/102 = 1/100 = 0.01.
Here is an example:
The logarithm of 375 (Log 375) is about 2.574. (Ten to the 2.574 power = 375.) This makes sense, for 10 to the 2nd power is 100, and 10 to the 3rd power is 1000. Thus, the exponent of ten, the logarithm for 375, must be somewhere between 2 and 3.
A logarithm consists of two parts.
In the case of the logarithm of 375, which is 2.574, the 2 is called the characteristic. The characteristic tells us how big the number is... like the words used to describe the characteristic of a person... He is a big fellow... or She is a petite lady. The number 2, the characteristic, tells us that the number is between 100 and 1000... between 10 to the 2nd power and 10 to the 3rd power.
The .574 is called the mantissa. (You find this on the L scale of your slide rule.) The mantissa .574 tells us where in that range the number actually is.
10 to the 2.574 power (102.574) = 375. Log 375=2.574
It is interesting to note that the mantissa of 375 is the same as the mantissa for 37.5 or 3.75 or 375,000,000! The characteristic tells us which number it is.
Log 3.75 = 0.574
Log 37.5 = 1.574
Log 375 = 2.574
Log 375,000,000 = 8.574
Each successive characteristic represents a number that is 10 times, or one tenth, the number represented by the adjacent characteristic. But ... logarithms of numbers less than 1 are slightly different! See end of this chapter.
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The basic scales, C and D, on a common 10 inch slide rule, are therefore 10 inch rulers whose markings are not in uniform inches, but in the number of inches of the mantissa for each number between zero and 10. The logarithm of 2 is 0.301... and you will find the number 2 on the C and D scales 3.01 inches from the left end. The logarithm of 3 is 0.477, and you will find the number 3 engraved 4.77 inches from the left end. Adding the two distances gives one 7.78 inches, and sure enough, the number 6 is 7.78 inches from the left end.
You dont actually have to add the distances to calculate the numbers. The slide rule has already done this for you.
To multiply 2 times 375, a slide rule adds the logarithms of the two numbers and reaches the logarithm of the answer. But... although the scales are spaced logarithmically, they are labeled with the actual numbers, not the logarithms. So, you read the answers directly.
2 (logarithm is 0.301) times 375 (logarithm is 2.574) = 750 (logarithm is 2.875) The logarithms are added. The numbers are multiplied.
For numbers < 1.0, we use a different approach to determine the logarithm. They are represented by a logarithm with a negative characteristic.
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To determine the logarithm of 0.375, we subtract 1 from the logarithm of 3.75.
Log 0.375 = Log 3.75 - 1.000 = 0.574 - 1.000 = -0.426.
We shifted the decimal point one place to the left of a single digit whole number (3.75), and in doing so we divided by a factor of 10, thus we subtract 1 from the logarithm of 3.75.
Another example:
Log 0.0375 = Log 3.75 - 2.000 = 0.574 - 2.000 = -1 .426
And ... Log 0.00375 = -2.426
This is the same process and reasoning we use for numbers greater than one. For example:
Log 37.5 = Log 3.75 + 1. Log 37.5 = 0.574 + 1 = 1.574.