==[]-- Short notation for slide rule algorithms

Throughout this site I use special notation to describe slide rule algorithms. This page describes this notation in detail and provides a tool to decode (translate) this notation into plain English.

Components and skills

Slide rule consists of a few important components. For successfull operation slide rule requires from its user particular skills in manipulating those components and general understanding how these components are built and how they work.

Scales on slide and body

Scales are the heart of rule. All other parts of slide rule either carry scales, make them easier to read, or change relationships between scales.

Scale is a line. Each point of a scale represents a real number. This number is sometimes called 'the name' of this point. Some points are marked with ticks and some ticks carry numeric and symbolic labels thus helping user to determine the name of the point. Distance between 2 ticks is choosen that way to allow linear approximation for points between them.

The formula that represents the distance from the begining of the scale to a point with name x is often caled "formula of a scale". Thus D scale has formula d = log10x (assuming that scale spans from 1 to 10 and its length is 1).

Moving slide

By moving slide you change algebraic relationship between slide and body scales. To be more specific -- if one of the scales is located on a slide had formula d=f(x), then after the slide has been moved l units right, then this scale has formula d=l+f(x).


Cursor is used to set, read and store numbers. The main part of cursor is its hairline (or hairlines) that simultaneously intersects several scales, thus giving correspondense between values on them.

What cursor essentially does, it finds the point on one scale that is on the same distance from the scale beginning as point on another scale.

Reading values

To read the number means "to determine the number represented by the given point of given scale". This point can be defined by cursor hairline, by tickmark or by tickmark on another scale that touches this point.

Setting values with cursor

This means to put cursor in position that its hairline passes through the desired point of the desired scale. If the value does not have tickmark then you have to interpolate between nearby tick marks.

Digit count (or characteristic)

Each point of most scales represent not single number, but the whole range of numbers that differ from each other by the power of ten factor. Deciding which particular factor (the number of digits in it is often called digit count of the number) correspond to actual value should be decided by person making calculations.

It is the toughest and the most important skill required for using slide rule.

Generally this document does not give rules to determine digit count. This topic is widely discussed in numerous slide rule manuals.



Manual calculations

Some algorithms reference operations that are easy for human but hard for slide rule. Example of such operation is addition: there is no easy way to perform it with typical slide rule.

What constitutes "easy for human" depends a lot on a specific human in question. It can be (in the order of increasing difficulty):

These steps are considered elementary though they can lead to big errors.


One very useful primitive operations is searching. Searching means the following:
For any given pair of scales (usually one is on the stock, another is on the slide) find the position of the cursor where numbers under hairline satisfy given relationship.

The most often occuring relationship is equality. In this case operation becomes "find such position of the cursor that it marks the same numbers for two given scales".

Searching with a slide

Sometimes more complicated case of searching than above may be useful. Put cursor into required position, then move slide, while monitoring the value of one of the slide located scales under cursor (a) and and a value on the stock scale (this scale should be adjacent to slide) pointed by slide index (or even by some constant, located on slide and adjacent to its edge) (b). When these a and b match the relation -- position of slide is found.

This type of search is used to calculate square roots without A or CI scale and also to calculate cubic roots without K scale. See (R. Manely site).

Short notation

Algorithms in this document are described in terms of elementary operations defined above. The notation used was designed to be short rather than clear.
Scales are denoted with capital latin letters: A, D, CI, K, ST etc.
Numbers are denoted either directly (using decimal notation) or with lowcase latin leters.

Input values of algorithm usualy get names u, v, w, t etc. Constants get names a, b, c etc. Values that should be found or intermediate values get names x, y, etc. Constants present on the scales denoted with their quoted names: 'pi', 'c1', 'rho'

Number followed by scale name represents point on the scale where this number is located.

scale index
1 (followed by scale name) denotes both left and right indices. Sometimes it is used even when 1 is not an actual mark on a left or right edge of the scale. When distinction between left and right is necessary [1 is used for left index and 1] is used for right.

Index does not necessarily means "left edge" or "right edge", since sometimes scales have extensions that extend beyond index. Example would be LL scales, S scale, ST scale etc.

Informally, index of a stock scale is number that is aligned with 1 or 10 on scale D and index of a slide scale is number that is aligned with 1 or 10 on scale C.

Denoted as |. Followed by scale name means the number under cursor on the scale.
Movements are denoted with an arrow: -> or <- or ←,→

There are two kinds of moving elements -- points on the scales located on the slide and cursor. Arrow points from moving entity to its position. Example: [1CI->| means "move left index of CI scale to cursor position".

Reading is denoted with bold arrow =>, <=, ⇐,⇒. Left part of arrow represents point under (or above) which reading should be done. Right part of the arrow defines scale where the result should be read from. If intermediate result was read then its name follows scale name (often in brackets).
manual calculations
Denoted by := sign. Left part indicates name of new (intermediate) value. Right part contains formula. Usually it is very simple formula, like y:=x+1.
searching with cursor
denoted as |->relation between scales. Means "move cursor into position where numbers on the appropriate scales under the cursor conform to the realtion". For example |->D=CI means "match the same number on D and CI scales".

Notation decoder

This experimental feature will allow you to translate algorithm written in slide rule notation into english. Most algorithms descriptions on this site are 'clickable' -- this brings the window with translation. However, if this is not working with your browser, you may try to enter the algorithm description in this form and click 'Translate' button.


http://www.howe.k12.ok.us/~jimaskew/askew/rules.htm employs different "table" notation.
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