Scales (Slide Rule)

The Faber-Castell 57/87 Rb Rietz rule, showing both scale names (on the left side) and self documenting descriptions (on the right side).

The primary means of calculation on all slide rules, a scale is a number line arranged in a way that facilitates certain operations when measured against other scales. Most scales are meant to multiply numbers together, accomplished by arranging the marks in a logarithmic layout, with further adjustments based on the particular operation desired. The oldest calculating tool to use such scales is the "Gunter rule" which had a single logarithmic scale from 1 to 10, which could be measured by calipers for the multiplication process. Almost all slide rules have at least four scales in common: A (B C) D, which will be explained below.

How to Read Scales

The operations of a slide rule are performed by adding physical lengths together between scales to find results. This is best illustrated through simple "linear" scales (such as those found on standard rulers, or the "number line" learned from elementary school). Taking two copies of the number line (of the same size) and laying them next to each other, it is possible to use the physical properties of the scale to add without any extra manipulation. Labeling the two scales "X" and "Y," we can observe an example:

The lower scale set at 2 on the upper scale, showing 3 opposite 5

2 + 3 = 5

On the first scale (X) we find the number 2. We can align the base of scale Y (called the "index") with it so that the 2 on X is next to 0 on Y. We then find the number 3 on scale X, and look for what number on scale Y is aligned with it (which is the number 5). Most operations on slide rules use this method to find various results.

Notation

There is no single standard for how to notate slide rule operations. Because of this, different sources have different means of describing actions. For the purposes of this wiki, the following standard will be used:

• All values will have the associated scale appended in parentheses. For example, the number 2.73 on scale D will be 2.73(D)
• Setting of two numbers adjacent to one another will be described by a vertical bar to represent their alignment. Aligning 23 on S with 11.9 on D will appear as 23(S) | 11.9(D)
• The index of all scales, with rare exception, is aligned on either end of the rule, and will be denoted as an uppercase I. If further clarification of scales is needed, it can be given; indices without a marked scale are assumed to be an index on the slide. Setting the index to 4 on D looks like I | 4(D)
• Steps will be separated out by an arrow →. Steps with only a single number imply setting the hairline to that number, steps which set two numbers together as above imply moving the slide.
• Not all numbers calculated are needed or recognized during steps (example given in a later section), any numbers used in calculation but not recognized will be denoted by simply naming the scale used in the calculation. e.g. setting 8.3(B) to a previously calculated value on D will look like 8.3(B) | (D)

Example

using the previous example of 2 + 3 = 5, we can write the actions as follows:

I(Y) | 2(X) → 3(Y) = 5(X)

which reads "Set the index of Y to 2 on X, then under 3 on Y read the answer 5 on X."

Using a Real Rule

A Slide rule consists of several scales arranged adjacent to one another, which can be quite some distance apart. To keep things aligned, a cursor with a "hairline" is provided which shows the vertical alignment of all scales on a given side. This allows for instantaneous transfer of values from one scale to another, which performs an operation dependent upon the orientation of the scales. This, combined with using the scales on the slide and body facilitate many complex operations with surprisingly few steps.

Unless otherwise designated, all scales on a slide rule are logarithmic, meaning that adding two lengths x and y on the scale will give you the product of the labeled lengths (xy). The following methods show how this can be used to perform most operations.

Off Scale

During the course of these operations, sometimes the desired result will fall "off" one end of a scale. This is only a minor difficulty, as the end of many scales either acts as an index in a wrap-around fashion (like the C scale) or it can get picked up on another scale to continue the operation (as in the LL scales, to be shown below). Sometimes this will necessitate a "change in index" where the bottom of the scale is replaced by the top of the same scale. This usually only happens in cases where the index was originally used in the calculation as a starting point, rather than an ending point.

Multiplication

The C and D scales are the primary scales against which most other scales are used. They are two identical scales in a logarithmic arrangement. Just like the above example shows how addition of two numbers can be achieved with linear scales, logarithmic scales can achieve multiplication the same way:

2 x 3 = 6
I | 2(D) → 3(C) = 6(D)

Notice that the process is the same: set the index of C to the first factor on D, find the second factor on C, and read the product below it back on D. Note that in some cases (like 9x2) a change of index may be required.

Division

Division is the inverse of subtraction, which means we can simply apply our process in reverse to do division.

3/2 = 1.5
3(D) | 2(C) → I = 1.5(D)

In some ways, the process for division is easier than for multiplication, as the fraction literally appears under the hairline, with the answer under the index. Because both indices of the C scale wrap around to represent the same place, there will always be at least one index that is on scale, and therefore will never need a change of index.

Squares and Square Roots

Squares and square roots can be calculated using only the C and D scales with some practice, but are more easily performed with the D and A (or C and B) scales. The A and B scales are often called the "square scales" because they are half the length of the C and D scales, which results in any number on D having its square on A (and likewise C with B). This is because of the property of logarithms in which doubling the logarithm of a number (in this case having the scale grow twice as fast) results in the logarithm of its square. To make it more clear, here is an example:

3² = 9
3(D) = 9(A)

Notice how squaring does not require any steps beyond finding the value to be squared.

Square roots are similarly easy to find, simply by reversing the steps:

16(B) = 4(C)

This does come with one complication: Because the A and B scales are half the length, they have two miniature "copies" of C and D (often called "decades) placed end to end. Because of this, one must take care which half of the scale to use when taking square roots. This can be found by counting the number of digits to the left of the decimal in the original number and using the lower decade for odd amounts of digits, and the upper decade for even amounts.

2.5(A) = 1.58(D)
25(A) = 5(D)

The K scale, common on many rules, performs cubes and cube roots (being a three decade scale) using the same process and a similar rule for amounts of digits.

Inverse Scales

In addition to A, B, C, and D, most slide rules have at least one "inverse" scale, which takes advantage of the fact that subtracting a physical length is the same as adding a length backwards. The most common scale is labeled "CI" and is a copy of the C scale reading right to left, instead of left to right. This allows for several operations to be simplified.

1/7 = 1.43...
7(C) ≈ 1.43(CI)

Notice that this can be set up in either direction, as performing a reciprocal operation is its own inverse.

Multiplication via Division

When multiplying two numbers, the rules of algebra tell us that AxB=A/(1/B). This means that we can perform our division process using the D and CI scales to achieve the same process as multiplying with C and D, but without any worry about going off scale.

9 x 2 = 18
9(D) | 2(CI) → I = 18(D)

This multiplication would normally have required using the upper index, which may not have been apparent from the outset, but with this method, the correct index is automatically set for us.

Division via Multiplication

through similar reasoning, we can perform our normal multiplication process to achieve a division using CI in place of C

8/2 = 4
I | 8(D) → 2(CI) = 4(D)

This gives us extra options for division to help find the most efficient method of complex problems.

Chaining Operations

As a single operation tool, the slide rule is already capable of speeding up calculation time. However, because of the analogue nature, each calculation has a margin of error. In order to both increase the power and accuracy of slide rule calculations, many operations can be "chained" together for complex procedures. Let's look at one such procedure that may require chaining.

Calculate the volume of a sphere with radius 2.48 using the following formula: V=4πr³/3

Notice here how we have a total of two multiplications, a division, a cube, and a special constant. This can most easily be accomplished on a rule with the following scales: C, D, CI, and a cube root scale, which here will be denoted R. With careful planning we can develop the following solution:

2.48(R) | 3(C) → 4(C) → (D) | π(CI) → I ≈ 63.8

Notice how the cubing and division are done together first. We can skip going to the index like we normally would, because we immediately need to multiply again, which starts by aligning the index to our number, which is already in place.

In general, chaining operations come in pairs that follow a pattern: divide then multiply. Align two numbers on scales, then move the hairline to the third number. Treating this new result as the first value of another chain, we can continue chaining values by alternating between moving the slide and hairline to tackle most complex products. Thanks to the inverse scales and how they interchange the multiplication and division processes, we can even perform long strings of multiplication or division in a much shorter set of steps. The only limit to this chaining ability is the user's creativity, familiarity with the tool, and the layout of the slide rule itself (which scales lie on the slide vs the body of the rule to facilitate the multiplications).

Decimal Points

Unlike modern keyed calculators, the slide rule does not (in most cases) give decimal points, instead relying on the user to place them correctly on their own. This allows for many scales to give multiple answers at once, or represent several operations at the same time. Users must practice properly placing the decimal point themselves, which can be helped through various methods.

Estimation

The most common method for simple calculations is to simply estimate. Many methods for estimation are taught, but a common one used in many American school systems is as follows: round all values to the largest significant digit, perform that calculation, and note the order of magnitude. The exact calculation is usually going to be the same order of magnitude. For example:

483 x 7.339 ≈ 500 x 7=3500
483(D) | I → 734(C) ≈ 354(D)
Therefore, 483 x 7.339 ≈ 3540 (3544.74 exact)

This method works very well for single calculations, or simple chains, but does not work well with extremely large or extremely small numbers, or long strings of numbers. For that, more advanced methods must be used.

Scientific Notation

A more secure method of estimation is to begin by writing every number in scientific notation, that is, of the form X x 10ⁿ where X is greater than (or equal to) 1, but less than 10. Slide rules are particularly adept at this, since numbers such as 2, 200, and 0.0002 are all at the same position on the slide rule, it is easier to think of them as simply 2 with some multiplier power of ten. Most slide rule instruction books write the positions on the actual scales in this form, with 346 on the rule being described as 3.46.

After all numbers are in the correct form, estimation can take place on the now one-digit numbers (called the mantissa), and simple addition and subtraction of the powers of ten (called the characteristic) can properly place the decimal.

431 x 0.00085 = (4.31x10²)x(8.5x10⁻⁴) ≈ 4x9x10²⁻⁴=36x10⁻²=0.36
4.31(D) | I → 8.5(C) ≈ 3.67(D)
therefore 431 x 0.00085 ≈ 0.367 (0.36635 exact)

Notice that it is not enough to add the characteristics, as the multiplication of the mantissas themselves can be two digits. This method tends to be much more secure and can be chained together with some practice, though multiple factors in a product or quotient can make tracking the extra digits of the mantissa more difficult.

Slide Counting

A peculiar feature of slide rules is the fact that the scales "wrap around" themselves quite often. This leads to a very nice tool in tracking any extra complications with the decimal point: the slide itself. When performing the standard multiplication procedure (starting at an index), any time the upper index is used, this represents using this "wrapping" feature, which in essence describes adding 1 to the characteristic at the end. With division, the operation is reversed, meaning that any time you reach the upper index, you have to subtract one from the characteristic. The actual rule takes more reasoning, but can be described as follows: if multiplying brings you down the rule, add one to the characteristic (using either method of multiplying). Likewise, if dividing (using either method) brings you up the rule, subtract one from the characteristic. This can be seen by using our previous example on scientific notation:

431 x 0.00085 = (4.31x10²) x (8.5x10⁻⁴) ⇒ 2 - 4 = -2
4.31(D) | I → 8.5(C) ≈ 3.67(D); (4 > 3) ⇒ Characteristic + 1 (=-1)
Therefore 3.67x10⁻¹ = 0.367

Notice here that we move down the rule from 4.31(D) to 3.67(D), which prompts us to add 1 to the characteristic.

Digit Tracking

An alternative to scientific notation which can sometimes be shown on slide rules presents the same ideas in a different form, using the fact that the order of magnitude of a product is either the sum of the orders of the factors, or one less than that. This essentially presents the same method as the scientific notation method, but reversing the rules: When the product up the rule from the multiplicand, subtract 1 from the sum of the magnitudes, and when a quotient is down the rule from the dividend, add one to the difference of magnitudes. This method ends up using the exact same mathematical principles, but from a point of view that can sometimes be more intuitive to users, as it doesn't require remembering that single digit numbers have a characteristic of 0.

Folded and Split Scales

Similar to how the A and B scales are the C and D scales shrunk down to fit "two decades" on the rule, other rules may have similar transformations that allow them to cover different ranges or have other nice properties. The general name for these scales are "folded" scales, because they "fold" up to fit onto the rule in the space provided. These folded scales come in two general forms: simple folded scales, and "split" scales.

Folded Scales

A simple folded scale takes advantage of the "wrapping around" property of the C and D scales, usually being a copy of them that is shifted to align the index to a different point on the scale. The most common of these scales are the CF and DF scales, which are traditionally folded at the value of π ≈ 3.1416 and only have a single index near the middle of the rule. This is advantageous for two reasons:

First, pi is very close to the square root of 10, which is the value at the exact middle of the C and D scales, meaning that values normally very far apart on C and D (like 9 and 15) are brought much closer together and can facilitate easier chaining of operations without having to perform changes of index.

Second, moving from one of the primary scale to its associated folded scale has the effect of multiplying the value by pi, which can lead to more accurate results involving pi that remove the potential error involved with extra slide manipulations.

Other common points to fold the scales are at √10 ≈ 3.16, log(e) ≈ 2.301, or ln(10) ≈ 0.434, each with their own advantages and disadvantages, depending on the layout of the rule

Split Scales

Instead of shrinking to fit multiple copies of a scale on a single line, the reverse process is also possible. Scales that have been cut or split and placed on multiple lines help to improve the resolution and accuracy of the rule without making the rule unreasonably large. The most common scales with this feature are the Log-Log scales, commonly abbreviated to LL. The LL scales are related to the C and D scales such that for any number found on the LL scales, the natural (or, rarely, the common) logarithm of that number is found on D. The "Log-Log" name comes from the fact that the C and D scales are already logarithmic, meaning the LL scales are doubly so. Other split scales include sine and tangent scales, which usually have their lowest range combined into a single "ST" scale; square root scales, sometimes called "W" scales, which are twice as long as D; and cube root scales, which are three times as long as D, and are most common on high end Pickett brand rules.

LL Scales

The LL scales are unique in that they do not wrap around on themselves, and instead have a set decimal place present directly on the scale. They are most commonly seen on Duplex style rules, with one notable exception being Darmstadt system rules, which also uniquely have LL scales on the slide rather than the base of the rule. LL scales usually come in groups of either three (LL 1,2,3 and their inverses) or in groups of four (LL 0,1,2,3). In both cases, LL3 is considered the "base" scale, with its lower index showing the base of the logarithmic relationship it has -- usually e, though in rare cases 10. The LL scales are set up in such a way that exponents can be easily found for any number. This is because any arbitrary exponent x^y = alog( y log(x) ), which just becomes a multiplication read from the LL scales. A good example here is as follows:

5⁴ = 625
5(LL3) | I → 4(C) = 625(LL3)

Because the LL scale is split up into four sections, it reads from e^0.001 up to e¹⁰, which is a range of about 1.001 to 20000. This leads to some accuracy problems at the high end, but on the low end it becomes incredibly accurate. One interesting thing to note is that for sufficiently small values of x (such as those less than 0.001), we have the approximation e^x ≈ 1+x, which means that for values below LL0, the C or D scale itself can act as a sort of "LL00" by showing the fractional portion of the value.

List of Common Scales

Below is a list of the common scales used on this page, as well as a few others. For a more complete list, see the List of Scales and Scale Marks

Common Scales

scale	relation to primary scales	Common Location	Notes
D	Primary scale (x)	body, directly adjacent to C	This is the most commonly used scale on the body of most slide rules. it is identical to C
C	Primary scale (x)	Slide, directly adjacent to D	This is the most commonly used scale overall. Often found on both sides of duplex rules
CI	Reciprocal (1/x)	Slide, usually adjacent to C	identical to C, but increasing from right to left. incredibly common, and very helpful in chaining operations together
A	Square, "quadratic" (x²)	Body, directly adjacent to B, usually above C and D	two decade scale, squares are performed D → A, square roots are performed A → D
B	Square, "quadratic" (x²)	Slide, directly adjacent to A	two decade scale, it is identical to A
K	Cubic scale (x³)	Body, in various locations, usually on the same side as A and B for duplex rules	three decade scale, one of the only purely logarithmic scales without a slide counterpart
DF	Folded scale (πx)	Body, usually on the opposite side of A and B, directly adjacent to CF, not common on simplex rules	identical to D, shifted by a constant. only has one index near the center of the scale. Can be used to "rescue" operations that otherwise need a change of indices. Useful for finding areas and circumferences of circles.
CF	Folded scale (πx)	Slide, directly adjacent to DF.	identical to DF. This scale can be used in place of C because of its single index on rules without scale extensions.
CIF	Folded Reciprocal (1/πx)	Slide, usually adjacent to CF	Identical to CF, but increasing from right to left. Can be used with CF and DF to chain operations similar to C, D, and CI. Not found on rules without other folded scales.
S	Sine scale (arcsin(x))	either body or slide	nearly always in degrees, both DMS or decimal. Usually used by reading sin(θ) on the scale, then using the equivalent value on C or D for other operations. Usually reads from 5.7° (arcsin(0.1)) to 90°. Cosine values are often printed on the same line, increasing from right to left.
T	Tangent scale (arctan(x))	either body or slide, usually with S	Same function as S scale, but for tangents. Usually has range from 5.7° to 45°. Values greater than 45° can be read by taking the complement of the value, occasionally this is marked either on the same scale or on a "T2" scale.
ST	Small Angle scale (arcsin(x)≈arctan(x))	usually between S and T	uses the approximation that for sufficiently small angles, sin(θ)=tan(θ)=θ when measured in radians. Used to find very small angles, and convert small angles in degrees to radians. has a range from 0.57° to 5.7°, below this limit, the ST scale effectively wraps on itself, giving arbitrarily small angles.
LL	Log-Log scale (ex)	Body, found on the slide of Darmstadt rules	Usually used with C to find arbitrary bases and exponents. LL is split into 3 (sometimes 4) scales, labeled LL1, LL2, LL3, sometimes LL0. Each portion has a range of ecx from 1≤x≤10, where the constant C is 1/1000, 1/100, 1/10, or 1, in ascending index order.
LL/	Reciprocal Log-Log scale (e-x)	Body, often found opposite LL scales	reciprocal of the LL scales, increases from right to left, using the negative exponents of the same divisions as the LL scales.