Slide rule scale

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Short description: Graduated markings, generally logarithmic, on slide rule
Keuffel and Esser 7" slide rule (5" scale, 1954)[1]

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 Before they were superseded by electronic calculators in the 1970s, slide rules were an important type of portable calculating instrument.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[note 1] 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.[2] 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[note 2] 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.[3]

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.[note 3] 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.[6][7]

Scales

Front of slide rule with multiple scales
Back of slide rule with multiple scales
Front and back of Aristo 0972 HyperLog duplex rule (1973)

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.[8] 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.[8][3]

Advanced slide rules have many scales and they are often designed with particular types of user in mind, for example electrical engineers or surveyors.[9][10] There are rarely scales for addition and subtraction but a workaround is possible.[note 4][11] The rule illustrated is an Aristo 0972 HyperLog, which has 31 scales.[note 5] The scales in the table below are those appropriate for general mathematical use rather than for specific professions.

Slide rule scales[8][14]
Label formula scale type range of x range on scale numerical range (approx) Increase / decrease[note 6] 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 x2 square 1 to 10 1 to 100 1 to 100 increase On body. Two log cycles at half the scale of C/D.[15][note 7]
B x2 square 1 to 10 1 to 100 1 to 100 increase On slider. Two log cycles at half the scale of C/D.[15][note 7]
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-sinh2(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[15]
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[15]
K x3 cube 1 to 10 1 to 103 1 to 1000 increase Three cycles at one third the scale of D[15]
L, Lg or M log10x Mantissa of log10 1 to 10 0 to 1.0 0 to 1.0 increase hence a linear scale
LL0 e0.001x log-log 1 to 10 e0.001 to e0.01 1.001 to 1.010 increase
LL1 e0.01x log-log 1 to 10 e0.01 to e0.1 1.010 to 1.105 increase
LL2 e0.1x log-log 1 to 10 e0.1 to e 1.105 to 2.718 increase
LL3, LL or E ex log-log 1 to 10 e to e10 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-x2) Pythagorean[note 8] 0.1 to 1.0 √(1-0.12) to 0 0.995 to 0 decrease calculating cosine from sine at small angles (ST)
H1 √(1+x2) Hyperbolic[note 8] 0.1 to 1.0 √(1+0.12) to √(1+1.02) 1.005 to 1.414 increase Set x on C or D scale.
H2 √(1+x2) Hyperbolic[note 8] 1 to 10 √(1+12) to √(1+102) 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-sinh2(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-sinh2(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

  1. 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.
  2. 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 log10 e to simplify conversion between base-10 and natural logarithms. When subscripted "/M", they fold at ln(10).
  3. 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.
  4. In slide rule terminology "log-log" means the scale is logarithmic applied over an inherently logarithmic scale.
  5. 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

Detail of some scale labels and 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 206.3x103 to find sine and tan of small angles[17]).[18][19] A cursor may have subsidiary hairlines beside the main one. For example, when one is over kilowatts the other indicates horsepower.[note 9][19][20] 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[19][21]
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)
' 0.785 π/4 ratio area of circle to diameter2
M 0.318 1/π reciprocal π
ρ, ρ0 or 1° 0.0175 π/180 radians per degree
R 57.29 180/π degrees per radian
ρ' 3.438x103 60x180/π arc minutes per radian[17]
ρ" 206.3x103 60x60x180/π arc seconds per radian[17]
c 2.154 [math]\displaystyle{ \sqrt[3]{10} }[/math] if no K scale
1n, L or U 2.303 1/log10e ratio loge to log10
N 1.341 HP per kW mechanical horsepower

Notes

  1. ↑ The body can also be called frame, base, stock or stator.
  2. ↑ A hairline is a very finely drawn line.
  3. ↑ To multiply two numbers, a and b, a point of the dividers is placed on the 1 marking and the dividers are adjusted so the other point is at a (or a multiple of 10 of a). Keeping the separation of the dividers fixed, one point is moved to b and the second point will indicate axb (or b/a if the second point is placed towards the 1 marking.[4][5]
  4. ↑ Note that (u+v)=v⋅(u/v+1) and (u-v)=v⋅(u/v-1) To implement this requires adding or subtracting 1 mentally.
  5. ↑ The Aristo 0952 HyperLog was being manufactured in 1973 and is 37.4 centimetres (14.7 in) in length overall with scales as follows. Front: LL00, LL01, LL02, LL03, DF (on the slider CF, CIF, L, CI, C) D, LL3, LL2, LL1 and LL00. Back: H2, Sh2, Th, K, A (on the slider B, T, ST, S, P, C) D, DI, Ch, Sh1, H1. Its gauge marks are π, ρ', ρ, e, 1/e, √2.[12][13]
  6. ↑ Whether annotations increase or decrease from left to right.
  7. ↑ 7.0 7.1 R1/R2 often easier to use for square root than A and B.[8]
  8. ↑ 8.0 8.1 8.2 See Savard for special considerations.[16]
  9. ↑ See image above of the back of the Aristo slide rule.

References

Citations

  1. ↑ "Slide Rule and Planimeter Sections". K&E Catalog 42nd Edition. Keuffel and Esser. 1954. p. 279. http://www.mccoys-kecatalogs.com/KECatalogs/1954/1954kecatp278.htm. Retrieved 25 June 2021. 
  2. ↑ Johnson (1949), Preface.
  3. ↑ 3.0 3.1 Savard, John J. G.. "Types of Slide Rules". Quadribloc. http://www.quadibloc.com/math/sr03.htm. 
  4. ↑ "Slide Rules". Hewlett Packard. https://www.hpmuseum.org/sliderul.htm. 
  5. ↑ Sangwin (2003), p. 4.
  6. ↑ Smith, David E. (1958). History of Mathematics, Vol. II. Dover Publications. p. 205. ISBN 9780486204307. 
    Stoll, Cliff (May 2006). "When Slide Rules Ruled". Scientific American 294 (5): 80–87. doi:10.1038/scientificamerican0506-80. PMID 16708492. Bibcode: 2006SciAm.294e..80S. 
    Cajori, Florian (1920). On the history of Gunter's scale and the slide rule during the seventeenth century. University of California press. https://archive.org/details/113597_001_009/page/n3/mode/2up. 
  7. ↑ Sangwin (2003).
  8. ↑ 8.0 8.1 8.2 8.3 Marcotte, Eric. "Types of Slide Rules and their Scales". https://www.sliderule.ca/scales.htm. 
  9. ↑ Johnson (1949), pp. 1–6.
  10. ↑ Johnson (1949), pp. 85,105–106,133–135,136–138,182–184,189–190.
  11. ↑ Nikitin, Andrey. "Addition and subtraction with slide rule". http://nsg.upor.net/slide/sradd.htm. 
  12. ↑ Seale, Steve K.. "Aristo 0972 Hyperlog". http://www.steves-sliderules.info/rule%20code/Aristo%200972.html. 
  13. ↑ Hamann, Christian M.. "Aristo - Hyperlog ( 25 cm scales )". http://public.beuth-hochschule.de/hamann/sliderules/arihyp.html. 
  14. ↑ Hamman, Christian-M. "The Principle of Slide Rules Appendix D". Beuth University of Applied Sciences. http://public.beuth-hochschule.de/hamann/sliderules. 
    "Illustrated Self-Guided Course On How To Use The Slide Rule". International Slide Rule Museum. https://sliderulemuseum.com/SR_Course.htm#Scales. 
    "Slide Rule Scales Page". Sphere Research. https://www.sphere.bc.ca/test/scales.html. 
  15. ↑ 15.0 15.1 15.2 15.3 15.4 Savard, John J. G.. "How Did a Slide Rule Work?". Quadribloc. http://www.quadibloc.com/math/slrint.htm. 
  16. ↑ Savard, John J. G.. "Special Scales". Quadribloc. http://www.quadibloc.com/math/sr02.htm. 
  17. ↑ 17.0 17.1 17.2 Manley, Ron. "Gauge points for small angles.". http://www.sliderules.info/a-to-z/trig/gaugepoints.htm. 
  18. ↑ Johnson (1949), pp. 144–145,219.
  19. ↑ 19.0 19.1 19.2 Seale, Steve K.. "Gauge marks". http://www.steves-sliderules.info/rule%20code/Gaugepoints.html. 
  20. ↑ Fernández, J.G. (30 April 2009). "Peripheral Hairlines in FaberCastell Cursors Layout and uses". https://slidetodoc.com/peripheral-hairlines-in-fabercastell-cursors-layout-and-uses/. 
    Manley, Ron. "Cursor hair lines". http://www.sliderules.info/a-to-z/cursor.htm. 
  21. ↑ Hamman, Christian-M.. "Slide Rules & Slide Calculators: (F) Marks on Slide Rules and their Meaning". http://public.beuth-hochschule.de/hamann/sliderules/srmain.html#88. 

Works cited

Further reading



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