# Slide rule scale

A slide rule scale is a line with graduated markings inscribed along the length of a slide rule used for mathematical calculations. The earliest such device had a single logarithmic scale for performing multiplication and division, but soon an improved technique was developed which involved two such scales sliding alongside each other. Later, multiple scales were provided with the most basic being logarithmic but with others graduated according to the mathematical function required.
Few slide rules have been designed for addition and subtraction, rather the main scales are used for multiplication and division and the other scales are for mathematical calculations involving trigonometric, exponential and, generally, transcendental functions. Before they were superseded by electronic calculators in the 1970s, slide rules were an important type of portable calculating instrument.

## Slide rule design

A slide rule consists of a body and a slider that can be slid along within the body and both of these have numerical scales inscribed on them. On duplex rules the body and/or the slider have scales on the back as well as the front. The slider's scales may be visible from the back or the slider may need to be slid right out and replaced facing the other way round. A cursor (also called runner or glass) containing one (or more) hairlines may be slid along the whole rule so that corresponding readings, front and back, can be taken from the various scales on the body and slider.

### History

In about 1620, Edmund Gunter introduced what is now known as Gunter's line as one element of the Gunter's sector he invented for mariners. The line, inscribed on wood, was a single logarithmic scale going from 1 to 100. It had no sliding parts but by using a pair of dividers it was possible to multiply and divide numbers. The form with a single logarithmic scale eventually developed into such instruments as Fuller's cylindrical slide rule. In about 1622, but not published until 1632, William Oughtred invented linear and circular slide rules which had two logarithmic scales that slid beside each other to perform calculations. In 1654 the linear design was developed into a wooden body within which a slider could be fitted and adjusted.

## Scales

Simple slide rules will have a C and D scale for multiplication and division, most likely an A and B for squares and square roots, and possibly CI and K for reciprocals and cubes. In the early days of slide rules few scales were provided and no labelling was necessary. However, gradually the number of scales tended to increase. Amédée Mannheim introduced the A, B, C and D labels in 1859 and, after that, manufacturers began to adopt a somewhat standardised, though idiosyncratic, system of labels so the various scales could be quickly identified.
Advanced slide rules have many scales and they are often designed with particular types of user in mind, for example electrical engineers or surveyors. There are rarely scales for addition and subtraction but a workaround is possible. The rule illustrated is an Aristo 0972 HyperLog, which has 31 scales. The scales in the table below are those appropriate for general mathematical use rather than for specific professions.
Slide rule scales :
| Label | formula | scale type | range of x | range on scale | numerical range (approx) | Increase / decrease | comment |
| C | x | fundamental scale | 1 to 10 | 1 to 10 | 1 to 10 | increase | On slider |
| D | x | fundamental scale used with C | 1 to 10 | 1 to 10 | 1 to 10 | increase | On body |
| A | x$^2$ | square | 1 to 10 | 1 to 100 | 1 to 100 | increase | On body. Two log cycles at half the scale of C/D. |
| B | x$^2$ | square | 1 to 10 | 1 to 100 | 1 to 100 | increase | On slider. Two log cycles at half the scale of C/D. |
| CF | x | C folded | π to 10π | π to 10π | 3.142 to 31.42 | increase | On slider |
| Ch | arccosh(x) | hyperbolic cosine | 1 to 10 | arccosh(1.0) to arccosh(10) | 0 to 2.993 | increase | note: cosh(x)= √(1-sinh$^2$(x)) (P) |
| CI | 1/x | reciprocal C | 1 to 10 | 1/0.1 to 1/1.0 | 10 to 1 | decrease | On slider. C scale in reverse direction |
| DF | x | D folded | π to 10π | π to 10π | 3.142 to 31.42 | increase | On body |
| DI | 1/x | reciprocal D | 1 to 10 | 1/0.1 to 1/1.0 | 10 to 1 | decrease | On body. D scale in reverse direction |
| K | x$^3$ | cube | 1 to 10 | 1 to 10$^3$ | 1 to 1000 | increase | Three cycles at one third the scale of D |
| L, Lg or M | log$_{10}$x | Mantissa of log$_{10}$ | 1 to 10 | 0 to 1.0 | 0 to 1.0 | increase | hence a linear scale |
| LL0 | e$^{0.001x}$ | log-log | 1 to 10 | e$^{0.001}$ to e$^{0.01}$ | 1.001 to 1.010 | increase | |
| LL1 | e$^{0.01x}$ | log-log | 1 to 10 | e$^{0.01}$ to e$^{0.1}$ | 1.010 to 1.105 | increase | |
| LL2 | e$^{0.1x}$ | log-log | 1 to 10 | e$^{0.1}$ to e | 1.105 to 2.718 | increase | |
| LL3, LL or E | e$^x$ | log-log | 1 to 10 | e to e$^{10}$ | 2.718 to 22026 | increase | |
| LL00 or LL/0 | e$^{-0.001x}$ | log-log | 1 to 10 | e$^{−0.001}$ to e$^{−0.01}$ | 0.999 to 0.990 | decrease | |
| LL01 or LL/1 | e$^{-0.01x}$ | log-log | 1 to 10 | e$^{−0.01}$ to e$^{−0.1}$ | 0.990 to 0.905 | decrease | |
| LL02 or LL/2 | e$^{-0.1x}$ | log-log | 1 to 10 | e$^{−0.1}$ to 1/e | 0.905 to 0.368 | decrease | |
| LL03 or LL/3 | e$^{−x}$ | log-log | 1 to 10 | 1/e to e$^{−10}$ | 0.368 to 0.00045 | decrease | |
| P | √(1-x$^2$) | Pythagorean | 0.1 to 1.0 | √(1-0.1$^2$) to 0 | 0.995 to 0 | decrease | calculating cosine from sine at small angles (ST) |
| H1 | √(1+x$^2$) | Hyperbolic | 0.1 to 1.0 | √(1+0.1$^2$) to √(1+1.0$^2$) | 1.005 to 1.414 | increase | Set x on C or D scale. |
| H2 | √(1+x$^2$) | Hyperbolic | 1 to 10 | √(1+1$^2$) to √(1+10$^2$) | 1.414 to 10.05 | increase | Set x on C or D scale. |
| R1, W1 or Sq1 | √x | square root | 1 to 10 | 1 to √10 | 1 to 3.162 | increase | for numbers with odd number of digits |
| R2, W2 or Sq2 | √x | square root | 10 to 100 | √10 to 10 | 3.162 to 10 | increase | for numbers with even number of digits |
| S | arcsin(x) | sine | 0.1 to 1 | arcsin(0.1) to arcsin(1.0) | 5.74° to 90° | increase and decrease (red) | also with reverse angles in red for cosine. See S scale in detail image. |
| Sh1 | arcsinh(x) | hyperbolic sine | 0.1 to 1.0 | arcsinh(0.1) to arcsinh(1.0) | 0.0998 to 0.881 | increase | note: cosh(x)= √(1-sinh$^2$(x)) (P) |
| Sh2 | arcsinh(x) | hyperbolic sine | 1 to 10 | arcsinh(1.0) to arcsinh(10) | 0.881 to 3.0 | increase | note: cosh(x)= √(1-sinh$^2$(x)) (P) |
| ST | arcsin(x) and arctan(x) | sine and tan of small angles | 0.01 to 0.1 | arcsin(0.01) to arcsin(0.1) | 0.573° to 5.73° | increase | also arctan of same x values |
| T, T1 or T3 | arctan(x) | tangent | 0.1 to 1.0 | arctan(0.1) to arctan(1.0) | 5.71° to 45° | increase | used with C or D. |
| T | arctan(x) | tangent | 1.0 to 10.0 | arctan(1.0) to arctan(10) | 45° to 84.3° | increase | Used with CI or DI. Also with reverse angles in red for cotangent. |
| T2 | arctan(x) | tangent | 1.0 to 10.0 | arctan(1.0) to arctan(10) | 45° to 84.3° | increase | used with C or D |
| Th | arctanh(x) | hyperbolic tangent | 1 to <10 | arctanh(0.1) to arctanh(1.0) | 0.1 to 3.0 | increase | used with C or D |

### Notes about table

* Some scales have high values at the left and low on the right. These are marked as "decrease" in the table above. On slide rules these are often inscribed in red rather than black or they may have arrows pointing left along the scale. See P and DI scales in detail image.
* In slide rule terminology, "folded" means a scale that starts and finishes at values offset from a power of 10. Often folded scales start at π but may be extended lengthways to, say, 3.0 and 35.0. Folded scales with the code subscripted with "M" start and finish at log$_{10}$ e to simplify conversion between base-10 and natural logarithms. When subscripted "/M", they fold at ln(10).
* For mathematical reasons some scales either stop short of or extend beyond the D = 1 and 10 points. For example, arctanh(x) approaches ∞ (infinity) as x approaches 1, so the scale stops short.
* In slide rule terminology "log-log" means the scale is logarithmic applied over an inherently logarithmic scale.
* Slide rule annotation generally ignores powers of 10. However, for some scales, such as log-log, decimal points are relevant and are likely to be marked.

## Gauge marks

Gauge marks are often added to the scales either marking important constants (e.g. π at 3.14159) or useful conversion coefficients (e.g. ρ" at 180*60*60/π or 206.3x10$^3$ to find sine and tan of small angles). A cursor may have subsidiary hairlines beside the main one. For example, when one is over kilowatts the other indicates horsepower. See π on the A and B scales and ρ" on the C scale in the detail image. The Aristo 0972 has multiple cursor hairlines on its reverse side, as shown in the image above.
Gauge marks :
| Symbol | value | function | purpose | comment |
| e | 2.718 | Euler's number | exponential functions | base of natural logarithms |
| π | 3.142 | π | | areas/volumes/circumferences of circles/cylinders |
| c or C | 1.128 | √(4/π) | | ratio diameter to √(area of circle) (different scales) |
| C' or C1 | 3.568 | √(40/π) | | ratio diameter to √(area of circle) (different scales) |
| ' | 0.785 | π/4 | | ratio area of circle to diameter$^2$ |
| M | 0.318 | 1/π | | reciprocal π |
| ρ, ρ$^0$ or 1° | 0.0175 | π/180 | radians per degree | |
| R | 57.29 | 180/π | degrees per radian | |
| ρ' | 3.438x10$^3$ | 60x180/π | arc minutes per radian | |
| ρ" | 206.3x10$^3$ | 60x60x180/π | arc seconds per radian | |
| c | 2.154 | ${\sqrt[{3}]{10}}$ | | if no K scale |
| 1n, L or U | 2.303 | 1/log$_{10}$e | ratio log$_e$ to log$_{10}$ | |
| N | 1.341 | | HP per kW | mechanical horsepower |