The S (sine) scale
This is a 2 section scale. The first section is usually for angles from about 5.8 degrees to 90 degrees. The second section, labeled ST, is for angles from about 0.58 degrees to about 5.8 degrees. See below for details of the ST scale.
Sine values are read on the C or D scales. Many sine scales are also labeled in reverse with red or other color numbers, which are, of course, the cosines. Cosine is the complement of sine, meaning Sin 30o= Cos 60o, or Sin X = Cos (90o X).
The scales on ST run from about 0.50 to 5.8 degrees.
The ST scale is for both sines and tangents, since they are very nearly the same for small angles.
Thus, on the ST scale, sin 4o = 0.0698 = tan 4o. Values are read on the C or D scales.
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The T (tangent) scales: Single Tangent Scale
The tangent scale is commonly found as a single scale on many rules ... and as a 2 section scale on many others.
In its simplest form, the graduations run from about 5.8 to 45 degrees. (Tangents of angles smaller than 5.8 degrees are found on the ST scale.) Tangent values are read on the C or D scales.
Double Tangent Scales
Some larger slide rules have two T scales on the body of the rule, often labeled T1 and T2. T1 runs from about 5.8 degrees to 45 degrees. T2 runs from 45 to about 84.3 degrees, at which point the value of the tangent becomes greater than 10 and is off the scale of the slide rule.
These scales often have the complementary angles labeled in red every ten degrees, running in reverse, which represent the cotangent values for these angles.
To find tangents using these scales you simply set the cursor on the angle and read the tangent directly on the D scale at the cursor. To multiply the resulting tangent by a number, using only one movement, use the CI scale.
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The L Scale
Remember that logarithms have 2 parts. the characteristic and the mantissa.
The L scale gives us the mantissa of the logarithm to the base 10 (the common logarithm), which tells us where in the specified range the number actually is.
Examples for logarithm of numbers greater than 1.
Examples for logarithm of numbers less than 1.
You may note that the L scale is linear, from 0 to 10, with even divisions just like a ruler. In some 10 scale designs, it can actually be used as a ruler, but its purpose is to convert the logarithmic distances of the C and D scales to their linear equivalent, which we then read as the logarithms mantissa.
We multiply and divide by adding and subtracting logarithmic distances on the A, B, C, D, CI, DI, CIF, CF and DF scales. In fact, we could also add and subtract using two L scales if they were present on the rule. The problem with this technique would be very limited dynamic range (perhaps 3 decades), but it is possible. Some childrens math aids are in fact made exactly this way, with two sliding linear scales (which to us are L scales) to add and subtract.
The LL Scales
The LL scales may be used to raise any number to any power ... or to extract any root from any number. Other uses are calculation of hyperbolic functions, exponential equations, compound interest and time credit payments. These are obviously very versatile scales.
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The LL scales are used with the D scale.
On large slide rules, there are 8 Log Log scales, covering numbers from 0.0005 to about 20,000. Only 3 scales appear on some smaller rules. Others have 6. Some have only one.
The numbers on the Log Log scales of most slide rules are based on natural logarithms. They represent powers of e, the base of natural logarithms, as related to the D scale.
e = 2.718281828...
Why such a strange number? e is a fundamental constant, like pi (p). It is defined as the area under a hyperbolic curve with certain boundaries.
The natural logarithm (designated by the abbreviation ln), with e as its base, is especially useful in calculus because its derivative is given by the simple equation d/dx ln x = 1/x. Also, e has the following unusual properties in calculus d/dx ex = ex = integral of ex dx. So, most of our slide rules have LL scales with the Base e because of calculus! Numbers on the D scale are logarithms, or powers, of numbers on the LL scales. Example: Cursor to x on LL, read natural logarithm of x (ln x) on D at cursor.
A few slide rules use common logarithms (Base 10) on their Log Log scales. On the well-known Pickett N4 you will read the common logarithm on D. Cursor to x on LL, read common logarithm of x [abbreviated as log x] on D.
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Scales LL0, LL1, LL2 and LL3 are called the log log scales. They represent positive powers of e. Scales LL/0, LL/1, LL/2 and LL/3 (or on some rules LL00, LL01, LL02, LL03) represent negative powers of e and are thus called reciprocal log log scales.
The relationship of successive adjacent log log scales is that of one-tenth powers of e. For example, if we set the cursor to 2 on D, we read e2 = 7.4 on LL3, e0.2 = 1.2215 on LL2, e0.02 = 1.0202 on LL1, e-2= 0.135 on LL/3, e-0.2 = 0.8187 on LL/2, etc.
Since e-x is the reciprocal of ex, any number on an LL scale has its reciprocal directly opposite on the corresponding reciprocal log log scale. In the example above, the reciprocal of 7.4 on LL3 is therefore 0.135 on LL/3 and the reciprocal of 1.2215 on LL2 is 0.8187 on LL/2, etc.
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We recommend the rough estimate method of determining the location of the decimal point. For example, if you are multiplying 402 X 0.071, the digits for the answer on your slide rule are 285. Rough estimate is 400 X 0.1 = 40. Answer is thus 28.5, not 2.85 or 285.
The following methods can also be used to locate the decimal point: