View source for Slide rule scale

{{Short description|Graduated markings, generally logarithmic, on slide rule}}
{{Use British English|date=June 2021}}
{{Use dmy dates|date=June 2021}}
[[File:Keuffel and Esser-Model 4181-1 Log log Duplex Decitrig slide rule-IMG 5821-white (cropped).jpg|thumb|upright=1.62|Keuffel and Esser 7" slide rule (5" scale, 1954)<ref>{{cite book |title=K&E Catalog 42nd Edition |date=1954 |publisher=Keuffel and Esser |page=279 |url=http://www.mccoys-kecatalogs.com/KECatalogs/1954/1954kecatp278.htm |chapter=Slide Rule and Planimeter Sections |access-date=25 June 2021 |archive-date=7 April 2021 |archive-url=https://web.archive.org/web/20210407131817/http://www.mccoys-kecatalogs.com/KECatalogs/1954/1954kecatp278.htm |url-status=live }}</ref>]]

A '''slide rule scale''' is a line with [[Graduation (scale)|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 functions|trigonometric]], [[exponential functions|exponential]] and, generally, [[transcendental function]]s. Before they were superseded by [[electronic calculator]]s in the 1970s, slide rules were an important type of portable calculating instrument.

==Slide rule design==
{{further|Slide rule#Physical design}}
A slide rule consists of a body{{refn|group=note|The body can also be called frame, base, stock or stator.}} and a slider that can be slid along within the body and both of these have numerical [[Level of measurement|scales]] inscribed on them. On duplex rules the body and/or the slider have scales on the back as well as the front.{{sfnp|Johnson|1949|loc=Preface}} 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 (slide rule)|cursor]] (also called runner or glass) containing one (or more) hairlines{{refn|group=note|A hairline is a very finely drawn line.}} 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.<ref name=savard />

=== 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.{{refn|group=note|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 {{math|''a''x''b''}} (or {{math|''b''/''a''}} if the second point is placed towards the 1 marking.<ref>{{cite web |title=Slide Rules |url=https://www.hpmuseum.org/sliderul.htm |website=Museum of HP Calculators |publisher=Hewlett Packard |access-date=25 June 2021 |archive-date=7 May 2021 |archive-url=https://web.archive.org/web/20210507055204/https://www.hpmuseum.org/sliderul.htm |url-status=live }}</ref>{{sfnp|Sangwin|2003|p=4}}}} 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.<ref>{{cite book |last1=Smith |first1=David E. |title=History of Mathematics, Vol. II |date=1958 |publisher=Dover Publications |isbn=9780486204307 |page=205}}<br />{{cite journal |last1=Stoll |first1=Cliff |title=When Slide Rules Ruled |journal=Scientific American |date=May 2006 |volume=294 |issue=5 |pages=80–87|doi=10.1038/scientificamerican0506-80 |pmid=16708492 |bibcode=2006SciAm.294e..80S }}<br />{{cite book |last1=Cajori |first1=Florian |title=On the history of Gunter's scale and the slide rule during the seventeenth century |date=1920 |publisher=University of California press |url=https://archive.org/details/113597_001_009/page/n3/mode/2up}}</ref>{{sfnp|Sangwin|2003}}

==Scales==
{{multiple image
 | direction = vertical
 | width = 360
 | image1 = Aristo 0972 HyperLog front.jpg
 | alt1 = Front of slide rule with multiple scales
 | image2 = Aristo 0972 HyperLog back.jpg
 | alt2 = Back of slide rule with multiple scales
 | footer = Front and back of Aristo 0972 HyperLog duplex rule (1973)
}}
Simple slide rules will have a C and D scale for [[multiplication]] and [[Division (mathematics)|division]], most likely an A and B for [[Square (algebra)|squares]] and [[square root]]s, and possibly CI and K for [[Multiplicative inverse|reciprocals]] and [[Cube (algebra)|cubes]].<ref name="erictypes">{{cite web |last1=Marcotte |first1=Eric |title=Types of Slide Rules and their Scales |url=https://www.sliderule.ca/scales.htm |website=Eric's Slide Rule Site |access-date=24 June 2021 |archive-date=31 March 2021 |archive-url=https://web.archive.org/web/20210331032653/http://www.sliderule.ca/scales.htm |url-status=live }}</ref> 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.<ref name="erictypes" /><ref name=savard>{{cite web |last1=Savard |first1=John J. G. |title=Types of Slide Rules |url=http://www.quadibloc.com/math/sr03.htm |website=www.quadibloc.com |publisher=Quadribloc |access-date=25 June 2021 |archive-date=28 September 2020 |archive-url=https://web.archive.org/web/20200928212105/http://www.quadibloc.com/math/sr03.htm |url-status=live }}</ref>

Advanced slide rules have many scales and they are often designed with particular types of user in mind, for example electrical engineers or surveyors.{{sfnp|Johnson|1949|pp=1–6}}{{sfnp|Johnson|1949|pp=85,105–106,133–135,136–138,182–184,189–190}} 
There are rarely scales for addition and subtraction but a workaround is possible.{{refn|group=note|Note that {{math|(''u''+''v''){{=}}''v''{{char|'''⋅'''}}(''u''/''v''+1)}} and {{math|(''u''-''v''){{=}}''v''{{char|'''⋅'''}}(''u''/''v''-1)}} To implement this requires adding or subtracting 1 mentally.}}<ref>{{cite web |last1=Nikitin |first1=Andrey |title=Addition and subtraction with slide rule |url=http://nsg.upor.net/slide/sradd.htm |website=nsg.upor.net |access-date=25 June 2021 |archive-date=11 November 2020 |archive-url=https://web.archive.org/web/20201111193530/http://www.nsg.upor.net/slide/sradd.htm |url-status=live }}</ref>
The rule illustrated is an Aristo 0972 HyperLog, which has 31 scales.{{refn|group=note|The Aristo 0952 HyperLog was being manufactured in 1973 and is {{convert|37.4|cm}} 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 {{strong|π}}, {{strong|ρ'}}, {{strong|ρ''}}, {{strong|e}}, {{strong|1/e}}, {{strong|√2}}.<ref name="stevearisto">{{cite web |last1=Seale |first1=Steve K. |title=Aristo 0972 Hyperlog |url=http://www.steves-sliderules.info/rule%20code/Aristo%200972.html |website=Steve's Slide Rules |access-date=24 June 2021 |archive-date=25 January 2020 |archive-url=https://web.archive.org/web/20200125184211/http://www.steves-sliderules.info/rule%20code/Aristo%200972.html |url-status=live }}</ref><ref>{{cite web |last1=Hamann |first1=Christian M. |title=Aristo - Hyperlog ( 25 cm scales ) |url=http://public.beuth-hochschule.de/hamann/sliderules/arihyp.html |website=public.beuth-hochschule.de |access-date=25 June 2021 |archive-date=4 April 2016 |archive-url=https://web.archive.org/web/20160404175313/http://public.beuth-hochschule.de/hamann/sliderules/arihyp.html |url-status=live }}</ref>}} The scales in the table below are those appropriate for general mathematical use rather than for specific professions.
{| class="wikitable sortable"
|+ Slide rule scales<ref name="erictypes" /><ref>{{cite web |last1=Hamman |first1=Christian-M |title=The Principle of Slide Rules Appendix D |url=http://public.beuth-hochschule.de/hamann/sliderules |publisher=Beuth University of Applied Sciences |access-date=25 June 2021 |archive-date=19 September 2020 |archive-url=https://web.archive.org/web/20200919122400/http://public.beuth-hochschule.de/hamann/sliderules/ |url-status=live }}<br />{{cite web |author1=International Slide Rule Museum |title=Illustrated Self-Guided Course On How To Use The Slide Rule |url=https://sliderulemuseum.com/SR_Course.htm#Scales |website=sliderulemuseum.com |publisher=International Slide Rule Museum |access-date=25 June 2021 |archive-date=23 May 2021 |archive-url=https://web.archive.org/web/20210523042758/https://www.sliderulemuseum.com/SR_Course.htm#Scales |url-status=live }}<br />{{cite web |author1=Sphere Research |title=Slide Rule Scales Page |url=https://www.sphere.bc.ca/test/scales.html |website=www.sphere.bc.ca |publisher=Sphere Research |access-date=25 June 2021 |archive-date=6 April 2021 |archive-url=https://web.archive.org/web/20210406220008/https://www.sphere.bc.ca/test/scales.html |url-status=live }}</ref>
|-
! Label 
! formula 
! class="unsortable" |scale type 
! class="unsortable" | range of ''x'' 
! class="unsortable" | range on scale 
! data-sort-type="number"| numerical range (approx)
! Increase / decrease{{refn|group=note|Whether annotations increase or decrease from left to right.}}
! class="unsortable"| comment
|-
| C || ''x'' || fundamental scale || 1 to 10 || 1 to 10 || data-sort-value="1000.2" | 1 to 10 || increase || On slider
|-
| D || ''x'' || fundamental scale used with C || 1 to 10 || 1 to 10 || data-sort-value="1000.2" | 1 to 10 || increase || On body
|-
| A || ''x<sup>2</sup>'' || [[Square (algebra)|square]] || 1 to 10 || 1 to 100 || data-sort-value="1000.3" | 1 to 100 || increase || On body. Two log cycles at half the scale of C/D.<ref name="savardhow">{{cite web |last1=Savard |first1=John J. G. |title=How Did a Slide Rule Work? |url=http://www.quadibloc.com/math/slrint.htm |website=www.quadibloc.com |publisher=Quadribloc |access-date=27 June 2021 |archive-date=10 October 2020 |archive-url=https://web.archive.org/web/20201010032114/http://www.quadibloc.com/math/slrint.htm |url-status=live }}</ref>{{refn|group=note|name=r1r2|R1/R2 often easier to use for [[square root]] than A and B.<ref name="erictypes" />}}
|-
| B || ''x<sup>2</sup>'' || [[Square (algebra)|square]] || 1 to 10 || 1 to 100 || data-sort-value="1000.3" | 1 to 100 || increase || On slider. Two log cycles at half the scale of C/D.<ref name="savardhow"/>{{refn|group=note|name=r1r2}}
|-
| CF || ''x'' || C folded || π to 10π || π to 10π  || data-sort-value="3142" | 3.142 to 31.42 || increase || On slider
|-
| Ch || arccosh(''x'') || [[hyperbolic cosine]] || 1 to 10 || arccosh(1.0) to arccosh(10) || data-sort-value="100" | 0 to 2.993 || increase || note: cosh(''x'')=<br>√(1-sinh<sup>2</sup>(''x'')) (P)
|-
| CI || 1/''x'' || [[Multiplicative inverse|reciprocal]] C || 1 to 10 || 1/0.1 to 1/1.0 || data-sort-value="10000" | 10 to 1 || decrease || On slider. C scale in reverse direction<ref name="savardhow" />
|-
| DF || ''x'' || D folded || π to 10π || π to 10π || data-sort-value="3142" | 3.142 to 31.42 || increase || On body
|-
| DI || 1/''x'' || [[Multiplicative inverse|reciprocal]] D || 1 to 10 || 1/0.1 to 1/1.0 || data-sort-value="10000" | 10 to 1 || decrease || On body. D scale in reverse direction<ref name="savardhow" />
|-
| K || ''x''<sup>3</sup> || [[Cube (algebra)|cube]] || 1 to 10 || 1 to 10<sup>3</sup> || data-sort-value="1000.4" | 1 to 1000 || increase || Three cycles at one third the scale of D<ref name="savardhow" />
|-
| L, Lg or M{{refn|group=note|name=Mskala|The Polish firm Skala had an "M" scale used in [[right triangle]] solutions.<ref name=Mskalaref>{{cite web | title = The "M" Scale in Polish Skala Slide Rules| first = Angel | last = Carrasco | translator1-last = González | translator-first1 = Alvaro | translator2-last = Fernández-Raventós | translator-first2 = José Gabriel | url = https://www.reglasdecalculo.com/presentaciones/escala_m/m_scale_uk.pdf | date = 2021-05-01 }}</ref>}} || log<sub>10</sub>''x'' || [[Mantissa (logarithm)|Mantissa]] of log<sub>10</sub> || 1 to 10 || 0 to 1.0 || data-sort-value="0" | 0 to 1.0 || increase || hence a [[linear]] scale
|-
| LL0 || ''e''<sup>0.001x</sup> || log-log || 1 to 10 || ''e''<sup>0.001</sup> to ''e''<sup>0.01</sup> || data-sort-value="1001" | 1.001 to 1.010 || increase ||
|-
| LL1 || ''e''<sup>0.01x</sup> || log-log || 1 to 10 || ''e''<sup>0.01</sup> to ''e''<sup>0.1</sup> || data-sort-value="1010" | 1.010 to 1.105 || increase ||
|-
| LL2 || ''e''<sup>0.1x</sup> || log-log || 1 to 10 || ''e''<sup>0.1</sup> to ''e'' || data-sort-value="1105" | 1.105 to 2.718 || increase ||
|-
| LL3, LL or E || ''e''<sup>''x''</sup> || log-log || 1 to 10 || ''e'' to ''e''<sup>10</sup> || data-sort-value="2718" | 2.718 to 22026 || increase ||
|-
| LL00 or LL/0 || ''e''<sup>-0.001x</sup> || log-log || 1 to 10 || ''e''<sup>−0.001</sup><br />to ''e''<sup>−0.01</sup> || data-sort-value="999" | 0.999 to 0.990 || decrease ||
|-
| LL01 or LL/1 || ''e''<sup>-0.01x</sup> || log-log || 1 to 10 || ''e''<sup>−0.01</sup><br />to ''e''<sup>−0.1</sup> || data-sort-value="990" | 0.990 to 0.905 || decrease ||
|-
| LL02 or LL/2 || ''e''<sup>-0.1x</sup> || log-log || 1 to 10 || ''e''<sup>−0.1</sup><br />to 1/''e'' || data-sort-value="905" | 0.905 to 0.368 || decrease ||
|-
| LL03 or LL/3 || ''e''<sup>−x</sup> || log-log || 1 to 10 || 1/''e''<br />to ''e''<sup>−10</sup> || data-sort-value="368" | 0.368 to 0.00045 || decrease ||
|-
| P || √(1-x<sup>2</sup>) || Pythagorean{{refn|group=note|name=savardsp|See Savard for special considerations.<ref name=savardspecial>{{cite web |last1=Savard |first1=John J. G. |title=Special Scales |url=http://www.quadibloc.com/math/sr02.htm |website=www.quadibloc.com |publisher=Quadribloc |access-date=27 June 2021 |archive-date=13 June 2021 |archive-url=https://web.archive.org/web/20210613151701/http://www.quadibloc.com/math/sr02.htm |url-status=live }}</ref>}} || 0.1 to 1.0 || √(1-0.1<sup>2</sup>) to 0 || data-sort-value="995" | 0.995 to 0|| decrease || calculating [[cosine]] from [[sine]] at small angles (ST)
|-
| H1 || √(1+x<sup>2</sup>) || Hyperbolic{{refn|group=note|name=savardsp}} || 0.1 to 1.0 || √(1+0.1<sup>2</sup>) to √(1+1.0<sup>2</sup>) || data-sort-value="1005" | 1.005 to 1.414|| increase  || Set x on C or D scale.
|-
| H2 || √(1+x<sup>2</sup>) || Hyperbolic{{refn|group=note|name=savardsp}}|| 1 to 10 || √(1+1<sup>2</sup>) to √(1+10<sup>2</sup>) || data-sort-value="1414" | 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 || data-sort-value="1000.1" | 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 || data-sort-value="3162" | 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) || data-sort-value="100" | 5.74° to 90° || increase and decrease&nbsp;(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) || data-sort-value="100" | 0.0998 to 0.881 || increase || note: cosh(''x'')=<br>√(1-sinh<sup>2</sup>(''x'')) (P)
|-
| Sh2 || arcsinh(''x'') || [[hyperbolic sine]] || 1 to 10 || arcsinh(1.0) to arcsinh(10) || data-sort-value="1000.2" | 0.881 to 3.0 || increase || note: cosh(''x'')=<br>√(1-sinh<sup>2</sup>(''x'')) (P)
|-
| ST || arcsin(''x'') and arctan(''x'') || [[sine]] and [[Tangent|tan]] of small angles|| 0.01 to 0.1 || arcsin(0.01) to arcsin(0.1) || data-sort-value="10" | 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) || data-sort-value="100" | 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) || data-sort-value="1000.2" |  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) || data-sort-value="1000.2" | 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) || data-sort-value="100" | 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 [[pi|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<sub>10</sub> ''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 scale|logarithmic]] applied over an inherently logarithmic scale.
# Slide rule annotation generally ignores [[power of 10|powers of 10]]. However, for some scales, such as log-log, decimal points are relevant and are likely to be marked.

==Gauge marks==
[[File:Aristo 0972 HyperLog (detail of labels).jpg|thumb|upright=1.6|Detail of some scale labels and gauge marks]]
Gauge marks are often added to the scales either marking important constants (e.g. {{strong|[[pi|π]]}} at 3.14159) or useful conversion coefficients (e.g. {{strong|ρ[[Quotation marks in English|"]]}} at 180*60*60/π or {{nowrap|206.3x10<sup>3</sup>}} to find sine and tan of small angles<ref name="manley" />).{{sfnp|Johnson|1949|pp=144–145,219}}<ref name="gaugemarks" /> A cursor may have subsidiary hairlines beside the main one. For example, when one is over kilowatts the other indicates horsepower.{{refn|group=note|See image above of the back of the Aristo slide rule.}}<ref name="gaugemarks">{{cite web |last1=Seale |first1=Steve K. |title=Gauge marks |url=http://www.steves-sliderules.info/rule%20code/Gaugepoints.html |website=Steve's Slide Rules |access-date=23 June 2021 |archive-date=21 January 2020 |archive-url=https://web.archive.org/web/20200121224604/http://www.steves-sliderules.info/rule%20code/Gaugepoints.html |url-status=live }}</ref><ref>{{cite web |last1=Fernández |first1=J.G. |title=Peripheral Hairlines in FaberCastell Cursors Layout and uses |url=https://slidetodoc.com/peripheral-hairlines-in-fabercastell-cursors-layout-and-uses/ |website=slidetodoc.com |access-date=25 June 2021 |date=30 April 2009 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625195426/https://slidetodoc.com/peripheral-hairlines-in-fabercastell-cursors-layout-and-uses/ |url-status=live }}<br />{{cite web |last1=Manley |first1=Ron |title=Cursor hair lines |url=http://www.sliderules.info/a-to-z/cursor.htm |website=www.sliderules.info |access-date=26 June 2021 |archive-date=18 March 2021 |archive-url=https://web.archive.org/web/20210318185253/http://www.sliderules.info/a-to-z/cursor.htm |url-status=live }}</ref>  See {{strong|π}} on the A and B scales and {{strong|ρ"}} on the C scale in the detail image. The Aristo 0972 has multiple cursor hairlines on its reverse side, as shown in the [[#Scales|image above]].

{| class="wikitable sortable"
|+ Gauge marks<ref name="gaugemarks" /><ref name=hamman>{{cite web |last1=Hamman |first1=Christian-M. |title=Slide Rules & Slide Calculators: (F) Marks on Slide Rules and their Meaning |url=http://public.beuth-hochschule.de/hamann/sliderules/srmain.html#88 |website=Pre-Computer Technical Museum |access-date=24 June 2021 |archive-date=4 January 2021 |archive-url=https://web.archive.org/web/20210104123446/http://public.beuth-hochschule.de/hamann/sliderules/srmain.html#88 |url-status=live }}</ref>
|-
! Symbol 
! data-sort-type="number"| value 
!class="unsortable" | function 
!class="unsortable" | purpose 
!class="unsortable" | comment
|-
| {{strong|e}} ||data-sort-value="271"| 2.718  || [[e (mathematical constant)|Euler's number]] || [[exponential functions]] || base of [[natural logarithm]]s
|-
| {{strong|π}} ||data-sort-value="314"| 3.142 || [[Pi|π]] ||  || areas/volumes/circumferences of circles/cylinders
|-
| {{strong|c}} or {{strong|C}} ||data-sort-value="112"| 1.128 || √(4/π) ||  
|rowspan=2| ratio diameter to √(area of circle) (different scales)
|-
| {{strong|C'}} or {{strong|C1}} ||data-sort-value="357"| 3.568 || √(40/π) ||  ||
|-
| {{strong|'}} ||data-sort-value="78"|0.785  || π/4 ||  || ratio area of circle to diameter<sup>2</sup>
|-
| {{strong|M}} ||data-sort-value="32"|0.318  || {{math|1/π}} ||  || reciprocal π
|-
| {{strong|ρ}}, {{strong|ρ<sup>0</sup>}} or {{strong|1°}} ||data-sort-value="02"| 0.0175  || {{math|π/180}} || [[radian]]s per [[degree (angle)|degree]] ||
|-
| {{strong|R}} ||data-sort-value="5729"| 57.29 || {{math|180/π}} || [[degree (angle)|degree]]s per [[radian]] ||
|-
| {{strong|ρ'}} ||data-sort-value="343800"| {{nowrap|3.438x10<sup>3</sup>}} || {{math|60x180/π}} || [[arc minute]]s per [[radian]]<ref name="manley">{{cite web |last1=Manley |first1=Ron |title=Gauge points for small angles. |url=http://www.sliderules.info/a-to-z/trig/gaugepoints.htm |website=www.sliderules.info |access-date=26 June 2021 |archive-date=18 March 2021 |archive-url=https://web.archive.org/web/20210318192130/http://www.sliderules.info/a-to-z/trig/gaugepoints.htm |url-status=live }}</ref> ||
|-
| {{strong|ρ"}} ||data-sort-value="20630000"| {{nowrap|206.3x10<sup>3</sup>}} || {{math|60x60x180/π}} || [[arc second]]s per [[radian]]<ref name="manley" /> ||
|-
| {{strong|c}} ||data-sort-value="215"| 2.154 || <math>\sqrt[3]{10}</math> || || if no K scale
|-
| {{strong|1n}}, {{strong|L}} or {{strong|U}} ||data-sort-value="230"| 2.303 || {{math|1/log<sub>10</sub>e}} || ratio [[Natural logarithm|log<sub>e</sub>]] to [[Common logarithm|log<sub>10</sub>]] ||
|-
| {{strong|N}} ||data-sort-value="134"| 1.341 || || [[Horsepower|HP]] per  [[kW]] || mechanical horsepower
|}

==Notes==
{{reflist|group=note}}

==References==
===Citations===
{{reflist}}

===Works cited===
{{refbegin}}
*{{cite book | last1 = Johnson | first1 = Lee Harnie | author-link1 =  | date = 1949 | chapter =  | script-chapter =  | trans-chapter =  | chapter-url =  | chapter-url-access =  | chapter-format =  | title = The Slide Rule | url = https://archive.org/details/sliderule00john | url-access = registration | language = en | publisher = [[David Van Nostrand|D. Van Nostrand]] | isbn =  | lccn = 49009467 | mr =  | oclc = 1450486 | ol = OL6049479M | access-date = 2022-06-14 | via = [[Internet Archive]] | df = dmy-all}}
*{{cite report | last1 = Sangwin | first1 = Christopher J. | title = Edmund Gunter and the Sector | publisher = [[University of Birmingham|School of Mathematics and Statistics, University of Birmingham School of Mathematics]] | date = 2003-01-21 | citeseerx = 10.1.1.524.1614 | s2cid = 204765145 |url = https://www.sliderulemuseum.com/Papers/EdmundGunterAndTheSector_ByCJSangwin.pdf |access-date = 2022-06-14 | df = dmy-all}}
{{refend}}

==Further reading==
*{{cite web |last1=Alfeld |first1=Peter |title=What Can You Do With A Slide Rule? |url=https://www.math.utah.edu/~alfeld/sliderules/ |website=www.math.utah.edu |publisher=University of Utah |access-date=25 June 2021|ref=none|archive-url=https://web.archive.org/web/20210625195930/https://www.math.utah.edu/~alfeld/sliderules/|archive-date=25 June 2021|url-status=live}}
*{{cite book |editor1-last=Davis |editor1-first=Richard |editor2-last=Hume |editor2-first=Ted |editor3-last=Koppany |editor3-first=Bob |title=Oughtred Society Slide Rule Reference Manual |date=2012 |publisher=Oughtred Society |url=http://www.oughtred.org/books/OSSlideRuleReferenceManualrevA.pdf|ref=none|archive-url=https://web.archive.org/web/20210426121808/http://www.oughtred.org/books/OSSlideRuleReferenceManualrevA.pdf|archive-date=26 April 2021|url-status=live}}
*{{cite book |last1=Harris |first1=Charles Overton |title=Slide Rule Simplified |date=1972 |publisher=American Technical Society |location=Chicago |isbn=978-0-8269-2342-4 |url=https://archive.org/details/sliderulesimplif0000harr/mode/2up|ref=none}}
*{{cite book |last1=Young |first1=Neville W. |title=A Complete Slide Rule Manual |date=1972 |publisher=David M. Peterson |url=https://www.qsl.net/ke4qdm/pdfmanual.htm|ref=none
|archive-url=https://web.archive.org/web/20210625200858/https://www.qsl.net/ke4qdm/pdfmanual.htm|archive-date=25 June 2021|url-status=live}}

{{Calculator navbox}}
{{DEFAULTSORT:Slide rule scale}}
[[Category:Analog computers]]
[[Category:Historical scientific instruments]]
[[Category:Mechanical calculators]]
[[Category:Logarithms]]
[[Category:Logarithmic scales of measurement]]