A slide rule scale is a line with
graduated markings inscribed along the length of a
slide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which ...
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 – hence the name slide rule (colloquially called a slipstick in the United States). 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
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
,
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
and, generally,
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
s. Before they were superseded by
electronic calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s 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 these have numerical
scales
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory), an object defined on a set of points
* Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original
* Scale factor, a number w ...
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
Cursor may refer to:
* Cursor (user interface), an indicator used to show the current position for user interaction on a computer monitor or other display device
* Cursor (databases), a control structure that enables traversal over the records in ...
(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.
In about 1620
Edmund Gunter
Edmund Gunter (158110 December 1626), was an English clergyman, mathematician, geometer and astronomer of Welsh descent. He is best remembered for his mathematical contributions which include the invention of the Gunter's chain, the Gunter's qu ...
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
The Fuller calculator, sometimes called Fuller's cylindrical slide rule, is a cylindrical slide rule with a helical main scale taking 50 turns around the cylinder. This creates an instrument of considerable precision – it is equivalent to a ...
. In about 1622, but not published until 1632,
William Oughtred
William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A General ...
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
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
, most likely an A and B for
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s, and possibly CI and K for
reciprocals and
cubes
In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
.
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
Victor Mayer Amédée Mannheim (17 July 1831 – 11 December 1906) was the inventor of the modern slide rule. Around 1850, he introduced a new scale system that used a ''runner'' to perform calculations. This type of slide rule became known u ...
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.
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
A power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The f ...
. Often folded scales start at
π but may be extended lengthways to, say, 3.0 and 35.0.
# 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. at 3.14159) or useful conversion coefficients (e.g. at 180*60*60/π or 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 on the A and B scales and 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.
Notes
References
Citations
Works cited
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Further reading
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{{DEFAULTSORT:Slide rule scale
Analog computers
Historical scientific instruments
Mechanical calculators
Logarithms
Logarithmic scales of measurement