List of Scales and Scale Marks

Slide rules consist of a set of analogue Scales which are set against one another to perform operations. Many scales exist and have different functions. Most slide rules also have marks that denote particular values that aid in calculations. Below is a list of scales and marks on many slide rules. While most common scales and marks have standardized names, rarer elements may not have such standards, and alternate names will be given.

Scales

The primary scales of a slide rule are C and D. unless otherwise specified, other scales are related to ("keyed" to) them. When showing relations, the primary scales will be described with a value of "x."

Most slide rules denote reversed scales (such as CI) in red. This tradition shall continue here for ease of reading which scales are ascending left to right, or are ascending right to left.

Scale name	Relation to primary scales	common location	Notes
D	Primary scale (x)	Body, directly adjacent to C	Original "Gunter Scales" only had one copy of what we now call the D scale, it was measured with calipers against itself, instead of against other scales.
C	Primary scale (x)	Slide, directly adjacent to D	The most commonly used scale on the slide rule. It is an exact copy of D.
CI	Reciprocal scale (1/x)	Slide, usually adjacent to C	Completes the complement of scales used in basic chaining of multiplication and division operations. exact copy of D running Right to Left
DI	Reciprocal scale (1/x)	Body, usually adjacent to D	an uncommon extra primary scale, can be used with CI the same as C with D. Gives extra flexibility for operations.
A	Square scale (x²)	Body, directly adjacent to B	"two decade" scale, progresses from 1 to 100 in one line. squares are read D → A, and roots are read A → D
B	Square scale (x²)	Slide, directly adjacent to A	Exact copy of A, allows for more flexibility in chaining operations.
BI	Reciprocal Square Scale (1/x²)	Slide, adjacent to B	Exact copy of A, running Right to Left. Unusual addition, increases flexibility of operations.
K	Cube scale (x³)	Body, usually same side as A and B on duplex rules	"three decade" scale, from 1 to 1000 in one line. Unlike A, does not usually have companion scales.
R (√, W/W')	Square root scale (√x)	Body, usually on the same side as A	the relationship R:C is the same as the relationship C:B, any numbers on R will have their associated square on C. This scale is necessarily split at √10 into two halves, usually right next to each other. On certain specialty rules which forego A and B, this scale is duplicated and set adjacent to each other just like C and D, which allow for the accuracy of a double length slide rule in a smaller package. In these cases, it is often called "W1/2" on the body and "W'1/2" on the slide. When used in conjunction with A or B, it can act to easily find 4th powers.
∛	Cube root scale (∛x)	Body, usually on the same side as K	The relationship ∛:C is the same as the relationship C:K, any number on the cube root scale will have its cube on C. This scale is necessarily split into three parts, at ∛10 and ∛100 respectively. Using this in combination with other power scales allow for various exponents, such as 6th powers or 9th powers with a single setting.
DF	Folded scale (πx)	Body, usually opposite side as A and B on duplex rules	Identical to D, shifted so that the index of D aligns with π on DF. Has effect of multiplying all values on D by π. Only has one index. Useful for calculations involving circles. Can also work with CF and CIF to "rescue" operations that would otherwise require a change of index.
CF	Folded scale (πx)	Directly adjacent to DF	Identical to DF, useful in performing operations near the high or low end of the scale, which are brought near the center of the scale.
CIF	Folded Reciprocal scale (1/πx)	Directly adjacent to CF	A reciprocal of CF, this completes the complement of folded scales, holding all functionality of D, C, and CI, including operating parallel to them to give better range for a given slide setting.
S	Sine scale (arcsin(x))	Lower body or center of slide (with other trig scales)	The first, and most common of the trig scales. Trig scales generally are read by finding the angle (in degrees) on the trig scale, then reading the associated value on C or D. The S scale has a range from 5.7° (0.1 radians) to 90°, either in DMS or decimal degrees. Sine (and cosine) is always less than or equal to 1, so all readings on C and D are the decimal portion.
Cos	cosine scale (arccos(x))	marked on same divisions as S	for convenience, S scales often have the complements of their values printed in red to allow for easy reading of cosine values.
T	Tangent scale (arctan(x))	Grouped with other trig scales	Range from 5.7° to 45°. In the absence of a second T scale, angles above this range can be calculated by taking tangent of the complement. Angles less than 45° are between 0 and 1, angles between 45° and 84.3° are between 1 and 10.
ST	Small Angle scale (arcsin(x)=arctan(x))	usually between S and T scales	This scale uses the approximation sin(x)=tan(x) for small angles. This is also approximately equal to the radian measure of the angle. This scale acts as an extension of both the S and T scales, with a range from 0.57° to 5.7°. Because of the approximation, this scale wraps back on itself for even smaller angles, decreasing the order of magnitude each time. tangents of angles near 90° can be found by finding the tangent of the complement.
T	Tangent scale (arctan(x))	marked to the same divisions as T	For convenience, many rules will write the complement of the values on the T scale to save some calculation time, this is best used with CI for direct reading.
T2	Tangent scale (arctan(x))	usually next to T	In some cases, a separate T scale from 45° to 84.3° will be provided which can be read with the C or D scales.
P	Pythagorean scale (√(1-x²))	often directly adjacent to S	Popular on European rules, most commonly the Darmstadt simplex layout. Allows for higher precision of both cosine and sine values in the more cramped range of the S scale. Functions as its own inverse, so can be used in flexible ways.
Sa (S)	Sine scale (arcsin(x²))	commonly on the back of the slide of Mannheim rules	A combination of S and ST which is keyed to the A scale, meaning values are read on A instead of C. This allowed for an extended low range that eliminated the need for an ST scale, at the cost of some precision in the high range.
Sh	Hyperbolic Sine scale (asinh(x))	uncommon, usually near LL scales	Set up such that reading the argument on Sh gives the result on C or D. Usually split into two scales adjacent to one another for higher accuracy. for an argument a > 3, sinh(a)≈ea/2, which can be found on the LL scales with the further approximation sinh(a)≈exp(a-0.7). Can be used with Th to calculate Cosh(x)
Th	Hyperbolic Tangent scale (atanh(x))	Usually adjacent to Sh	Often found in pairings with Sh. Similarly set up such that the argument read on Th has result on C or D. Can be used to calculate Cosh(x) with the formula sinh(x)/tanh(x)=cosh(x)