Naismith's rule

{{Short description|Rule of thumb for hiking time}} [[File:Naismith's rule.svg|thumb|320px|Naismith's rule<ref name="Naismith1892" /><ref name="Holman10" />]] '''Naismith's rule''' helps with the planning of a walking or hiking expedition by calculating how long it will take to travel the intended route, including any extra time taken when walking uphill. This rule of thumb was devised by William W. Naismith, a Scottish mountaineer, in 1892.<ref name="Naismith1892" /><ref name=Thompson2010/><ref name="Scarf08" /> A modern version can be formulated as follows: :Allow one hour for every {{convert|3|mi|km|sigfig=1}} forward, plus an additional hour for every {{convert|2000|ft|m|sigfig=1}} of ascent.<ref name="Holman10" /><ref name="Aitken1977" />

==Assumptions and calculations== [[File:Pace, Naismith's rule.svg|thumb|220px|Pace<ref name="Kay12_1" /> in minutes per kilometre or mile vs. slope angle resulting from Naismith's rule<ref name=Magyari2012/> for basal speeds of{{brk}}5{{nbs}}km/h and 4{{nbs}}km/h.{{refn|group=n|name=a|Speed and pace for the Naismith rule were calculated here for its metric version (5 kilometres horizontally and 600 meters of ascent), not the original one (3 mi and 2,000 ft).<br />In case of Naismith rule and Langmuir corrections the same, not modified value of ascent and descent was used for the distance of 4 km as for 5 km – 600 m for the Naismith rule and 300 m for Langmuir corrections (not taking into account the equivalence between distance and climb).}}]]

The original Naismith's rule from 1892 says that one should allow one hour per three miles on the map and an additional hour per 2000 feet of ascent.<ref name="Naismith1892" /><ref name="Scarf08" /> It is included in the last sentence of his report from a trip.<ref name="Naismith1892" /><ref name="MacInnes13" />

Today it is formulated in many ways. Naismith's 1 h / 3&nbsp;mi + 1 h / 2000&nbsp;ft can be replaced by: * 1 h / 3&nbsp;mi (5&nbsp;km) + 1 h / 2000&nbsp;ft (600 m)<ref name="Holman10" /><ref name="Aitken1977" /><ref name="MWC" /> * 1 h / 5&nbsp;km (3&nbsp;mi) + 1/2 h / 300&nbsp;m (1000&nbsp;ft)<ref name="Evans10" /><ref name="Marsh12" /><ref name="Bagshaw06" /> * 3 mph + ½ h / 1000&nbsp;ft<br />5&nbsp;km/h + ½ h / 300&nbsp;m<ref name=Langmuir2013 />{{refn|group=n|name=b|Langmuir 2013 recalls the Naismith's rule from 1892 in miles and feet, but further gives and uses it in metric system, climbing sometimes per contour line on a map (10 m or 50 m).<ref name=Langmuir2013 /> }} * 12 min / 1&nbsp;km + 10 min / 100 m<ref name="MacInnes13" />

The basic rule assumes hikers of reasonable fitness, on typical terrain, and under normal conditions. It does not account for delays, such as extended breaks for rest or sightseeing, or for navigational obstacles. For planning expeditions a team leader may use Naismith's rule in putting together a route card.{{Citation needed|date=January 2017}}

It is possible to apply adjustments or "corrections" for more challenging terrain, although it cannot be used for scrambling routes. In the grading system used in North America, Naismith's rule applies only to hikes rated Class 1 on the Yosemite Decimal System, and not to Class 2 or higher.{{Citation needed|date=January 2017}}

In practice, the results of Naismith's rule are usually considered the minimum time necessary to complete a route, though modern adaptations and hiking time calculators account for terrain difficulty, elevation gain, and individual fitness levels.<ref>{{cite journal |last=Langmuir |first=Eric |year=1995 |title=Mountaincraft and Leadership |publisher=SportScotland |page=23}}</ref>

When walking in groups, Naismith’s rule is generally applied based on the pace of the slowest member to ensure the group remains together. This adjustment accounts for variations in fitness, terrain difficulty, and rest needs among participants.<ref name="Langmuir2013" />

Naismith's rule appears in UK statute law, although not by name. The Adventure Activities Licensing Regulations apply to providers of various activities including trekking. Part of the definition of trekking is that it is over terrain on which it would take more than 30 minutes to reach a road or refuge (by the quickest safe route), based on a walking speed of 5 kilometres per hour plus an additional minute for every 10 metres of ascent.<ref name="legislationgovuk" />

[[File:Hiking speed.svg|thumb|none|400px|A plot of walking speed versus slope resulting from Naismith's rule<ref name=Magyari2012/> and Langmuir corrections<ref name=Magyari2012/><ref name=Langmuir1984/> for base speeds of 5&nbsp;km/h and 4&nbsp;km/h compared to Tobler's hiking function.<ref name=Tobler1993/>{{refn|group=n|name=a}}]]

==Scarf's equivalence between distance and climb== Alternatively, the rule can be used to determine the equivalent flat distance of a route. This is achieved by recognising that Naismith's rule implies an equivalence between distance and climb in time terms: 3 miles (=15,840 feet) of distance is equivalent in time terms to 2000 feet of climb.<ref name=Scarf2007/>

Professor Philip Scarf, Associate Dean of Research and Innovation and Professor of Applied Statistics at the University of Salford,<ref>{{cite web |url=http://www.salford.ac.uk/business-school/business-academics/philip-scarf |title=Professor Philip Scarf |author=<!--Not stated--> |website=www.salford.ac.uk |publisher=University of Salford |access-date= 1 February 2018 }}</ref> in research published in 2008, gives the following formula:<ref name="Scarf08" /> : equivalent distance = x + α·y where: :x = horizontal distance :y = vertical distance :α = 7.92 (3 mi / 2000 ft<ref name=Scarf2007 /><ref name="Scarf08" /><ref name="Kay12_2" />), called Naismith’s number by Scarf<ref name=Scarf2007 /><ref name="Scarf08" /><ref name="Kay12_2" />

That is, 7.92 units of distance are equivalent to 1 unit of climb. For convenience an 8 to 1 rule can be used. So, for example, if a route is {{convert|20|km}} with 1600 metres of climb (as is the case on leg 1 of the Bob Graham Round, Keswick to Threlkeld), the equivalent flat distance of this route is 20+(1.6×8)={{convert|32.8|km}}. Assuming an individual can maintain a speed on the flat of 5&nbsp;km/h, the route will take 6 hours and 34 minutes. The simplicity of this approach is that the time taken can be easily adjusted for an individual's own (chosen) speed on the flat; at 8&nbsp;km/h (flat speed) the route will take 4 hours and 6 minutes. The rule has been tested on fell running times and found to be reliable.<ref name=Scarf2007/> Scarf proposed this equivalence in 1998.<ref name="Scarf08" /><ref name="Kay12_1" />

As you can see the forward, the Scarf's assumption allows also to calculate the time for each speed, not just one as in case of the original Naismith rule.

===Pace=== Pace is the reciprocal of speed. It can be calculated here from the following formula:<ref name="Kay12_1" /><ref name="Kay12_2" />

:p = p0·(1 + α·m)

where: :p = pace :p0 = pace on flat terrain :m = gradient uphill

This formula is true for m≥0 (uphill or flat terrain).<ref name="Kay12_1" /><ref name="Kay12_2" /> It assumes equivalence of distance and climb by applying mentioned earlier α factor.<ref name="Scarf08" /><ref name="Kay12_2" />

Sample calculations: p0 = 12 min / km (for 5&nbsp;km / h speed), m = 0.6&nbsp;km climb / 5&nbsp;km distance = 0.12, p = 12 · (1 + 7.92 · 0.12) = 23.4 min / km.

==Other modifications== Over the years several adjustments have been formulated in an attempt to make the rule more accurate by accounting for further variables such as load carried, roughness of terrain, descents and fitness (or lack of it). The accuracy of some corrections is disputed,<ref name=Smith2009/> in particular the speed at which walkers descend a gentle gradient. No simple formula can encompass the full diversity of mountain conditions and individual abilities. ===Tranter's corrections=== {{Unreferenced section|date=September 2019}} Tranter's corrections make adjustments for fitness and fatigue. Fitness is determined by the time it takes to climb 1000 feet over a distance of ½ mile (800 m). Additional adjustments for uneven or unstable terrain or conditions can be estimated by dropping one or more fitness levels.

{|border="0" cellpadding="5" cellspacing="0" style="background:#ffffc0" |- |rowspan=2|Individual fitness in minutes |colspan=16 align="center"|Time taken in hours estimated using Naismith's rule |-style="background:#ffc080" |2||3||4||5||6||7||8||9||10||12||14||16||18||20||22||24 |- ! style="background:#ffc080"|15 (very fit) |1||1.5||2||2.75||3.5||4.5||5.5||6.75||7.75||10||12.5||14.5||17||19.5||22||24 |- ! style="background:#ffc080"|20 |1.25||2.25||3.25||4.5||5.5||6.5||7.75||8.75||10||12.5||15||17.5||20||23 |colspan=2 rowspan=3 style="background:silver"| |- ! style="background:#ffc080"|25 |1.5||3||4.25||5.5||7||8.5||10||11.5||13.25||15||17.5 |colspan=3 rowspan=2 style="background:silver"| |- ! style="background:#ffc080"|30 |2||3.5||5||6.75||8.5||10.5||12.5||14.5 |colspan=3 style="background:silver"| |- ! style="background:#ffc080"|40 |2.75||4.25||5.75||7.5||9.5||11.5 |colspan=2 rowspan=2 style="background:silver"| |colspan=8 align="center" style="background:silver; color:red"|Too much to be attempted |- ! style="background:#ffc080"|50 (unfit) |3.25||4.75||6.5||8.5 |colspan=12 style="background:silver"| |} For example, if Naismith's rule estimates a journey time of 9 hours and your fitness level is 25, you should allow 11.5 hours.

===Aitken corrections=== Aitken (1977) assumes that 1 h takes to cover 3&nbsp;mi (5&nbsp;km) on paths, tracks and roads, while this is reduced to 2½ mi (4&nbsp;km) on all other surfaces.<ref name=Aitken1977/>

For both distances he gives an additional 1 h per 2000&nbsp;ft (600 m) of ascent.<ref name=Aitken1977/> So Aitken doesn't take into account equivalence between distance and climb (proposed by Scarf in 1998<ref name="Scarf08" /><ref name="Kay12_1" />).

===Langmuir corrections===

Langmuir (1984) extends the rule on descent. He assumes the Naismith's base speed of 5&nbsp;km/h and makes the following further refinements for going downhill:<ref name=Langmuir2013 /><ref name=Langmuir1984 /><ref name=Caffin2013 /> * For a gentle decline (slopes between 5&nbsp;degrees and 12&nbsp;degrees) subtract 10 minutes for every 300&nbsp;meters of descent<ref name=Langmuir2013 /><ref name=Langmuir1984 /><ref name=Caffin2013 /> * For a steep decline (slopes greater than 12&nbsp;degrees) add 10 minutes for every 300&nbsp;meters of descent<ref name=Langmuir1984 /><ref name=Caffin2013 /> Later he says that the fitness of the slowest member of a party should be taken into account and thus a more practical formula for a group is:<ref name=Langmuir2013 /> * 4&nbsp;km/h + 1&nbsp;h / 450 m of ascent<ref name=Langmuir2013 />

==See also== * Preferred walking speed * Tobler's hiking function

==Notes== {{reflist|group=n}}

==References== {{reflist|30em|refs=

<ref name="Naismith1892">{{cite journal |last=Naismith |first=W. W. |date=September 1892 |title= Excursions. Cruach Ardran, Stobinian, and Ben More |url=http://gdl.cdlr.strath.ac.uk/smcj/smcj009/smcj0090603.htm |journal=Scottish Mountaineering Club Journal |volume=2 |issue=3 |pages=136 |access-date=22 January 2017 }} Available also in: [https://books.google.com/books?id=mTjuAgAAQBAJ&pg=PA136 Google Books]</ref>

<ref name=Aitken1977>{{cite thesis|last=Aitken|first=Robert|title=Wilderness Areas in Scotland, unpublished Ph.D. Thesis. University of Aberdeen. |url=http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.447113 |location=Aberdeen |year=1977 |access-date=26 January 2017|type=Ph.D }}</ref>

<ref name=Langmuir2013>{{cite book |last=Langmuir |first=Eric |title=Mountaincraft and Leadership; A Handbook for Mountaineers and Hillwalking Leaders in the British Isles |pages=38–39 |edition=Fourth |publisher=Mountain Training England; Mountain Training Scotland |year=2013 |isbn=978-0-9568869-0-3 }}</ref>

<ref name=Caffin2013>{{cite web |url=http://www.bushwalking.org.au/FAQ/FAQ_Navigation.htm|title=FAQ - Navigation: Walking Speed - Naismith's Rule |last=Caffin |first=Roger |access-date=23 March 2013}}</ref>

<ref name=Langmuir1984>{{cite book |last=Langmuir |first=Eric |title=Mountaincraft and Leadership. Official Handbook of the Mountain Leader Training Boards of Great Britain and Northern Ireland. |publisher=Britain & Scottish Sports Council |location=Edinburgh Scotland |year=1984}}</ref>

<ref name=Magyari2012>{{cite journal|last1=Magyari-Sáska|first1=Zsolt|last2=Dombay|first2=Ştefan|year=2012|title=Determining minimum hiking time using DEM|journal=Geographia Napocensis|volume=Anul VI|issue=2|pages=124–9|url=http://geographianapocensis.acad-cluj.ro/Revista/volume/nr_2_2012/pdf/Magyari_Dombay.pdf|access-date=21 March 2013}}</ref>

<ref name=Scarf2007>{{cite journal |last=Scarf |first=Philip |title=Route choice in mountain navigation, Naismith's rule, and the equivalence of distance and climb |journal=Journal of Sports Sciences |volume=25 |issue=6 |pages=719–726 |date=20 Mar 2007 |doi=10.1080/02640410600874906 |pmid=17454539 |s2cid=13897101 }} Also available in: [https://www.researchgate.net/publication/6373702_Route_Choice_in_Mountain_Navigation_Naismith's_Rule_and_the_Equivalence_of_Distance_and_Climb ResearchGate]</ref>

<ref name=Smith2009>[http://www.grough.co.uk/view/2009/07/13/just-a-minute-mr-naismith-can-that-be-right of downhill correction for Naismith's rule]</ref>

<ref name=Thompson2010>{{cite book|last=Thompson|first=S|title=Unjustifiable risk? The story of British climbing|edition=1st|chapter=1865–1914: gentlemen and gymnasts|pages=51–122|publisher=KHL Printing|location=Singapore|year=2010|isbn=978-1-85284-627-5 |chapter-url=https://books.google.com/books?id=mYW8LAHLNzoC&q=William+Naismith%2C+a+founder&pg=PA102}}</ref>

<ref name=Tobler1993>{{cite journal|last=Tobler|first=W|title=Three presentations on geographical analysis and modeling: Non-isotropic geographic modeling speculations on the geometry of geography global spatial analysis|journal=National Center for Geographic Information and Analysis Technical Report|volume=93|issue=1|pages=1–24|date=February 1993|url=http://www.ncgia.ucsb.edu/Publications/Tech_Reports/93/93-1.PDF|access-date=21 March 2013|archive-date=22 April 2008|archive-url=https://web.archive.org/web/20080422231635/http://www.ncgia.ucsb.edu/Publications/Tech_Reports/93/93-1.PDF|url-status=dead}} Available also in [http://www.geodyssey.com/papers/tobler93.html HTML] {{Webarchive|url=https://web.archive.org/web/20160304061057/http://www.geodyssey.com/papers/tobler93.html |date=2016-03-04 }} format.</ref>

<ref name="Kay12_1">{{cite journal |last=Kay |first=A. |title=Route Choice in Hilly Terrain |journal=Geogr Anal |volume=44 |issue=2 |pages=87–108 |date=2012 |url=http://www.lboro.ac.uk/microsites/maths/research/preprints/papers10/10-10.pdf |doi=10.1111/j.1538-4632.2012.00838.x |bibcode=2012GeoAn..44...87K |access-date=19 January 2017 |archive-url=https://wayback.archive-it.org/all/20121114225956/http://www.lboro.ac.uk/microsites/maths/research/preprints/papers10/10-10.pdf |archive-date=2012-11-14 |url-status=dead |citeseerx=10.1.1.391.1203 |s2cid=14054589 }}</ref>

<ref name="Kay12_2">{{cite journal |last=Kay |first=A. |title=Pace and critical gradient for hill runners: an analysis of race records |journal=Journal of Quantitative Analysis in Sports |volume=8 |issue=4 |date=November 2012 |url=https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/16478/1/PaceCG_published.pdf |doi=10.1515/1559-0410.1456 |s2cid=15045011 |issn=1559-0410 |access-date=19 January 2017}}</ref>

<ref name="Scarf08">{{cite journal |last=Scarf |first=Philip |title=A mathematical excursion in the isochronic hills |url=https://salford-repository.worktribe.com/output/1445747/a-mathematical-excursion-in-the-isochronic-hills|journal=Mathematics Today |volume=44 |pages=163–167 |date=August 2008 |access-date=22 January 2017}}</ref>

<ref name="Holman10">{{cite book |last=Holman |first=Tom |title=A Lake District Miscellany |publisher=Frances Lincoln |year=2010 |isbn=978-1907666384 |url=https://books.google.com/books?id=_inBAgAAQBAJ&pg=PT111 |access-date=19 January 2017 }}</ref>

<ref name="MacInnes13">{{cite book |last=MacInnes |first=Kellan |title=Caleb's List: Climbing the Scottish Mountains Visible from Arthur's Seat |publisher=Luath Press Ltd |year=2013 |isbn=978-1909912069 |url=https://books.google.com/books?id=-H_YBAAAQBAJ }}</ref>

<ref name="Evans10">{{cite book |last=Evans |first=Thammy |title=Macedonia; the Bradt Travel Guide |series=Bradt Guides |publisher=Bradt Travel Guides |year=2010 |isbn=978-1841622972 |url=https://books.google.com/books?id=YVPshBLnW_cC&pg=PA306 }}</ref>

<ref name="Marsh12">{{cite book |last=Marsh |first=Terry |title=Walking on the West Pennine Moors: 30 routes in gritstone country |publisher=Cicerone Press Limited |year=2012 |isbn=978-1849655392 |url=https://books.google.com/books?id=4zgEg1WxBqgC&pg=PT27 }}</ref>

<ref name="Bagshaw06">{{cite book |last=Bagshaw |first=Chris |title=The Ultimate Hiking Skills Handbook |publisher=David & Charles |year=2006 |isbn=978-0715322543 |url=https://books.google.com/books?id=EXpVicI5r1IC&pg=PA48 }} (5 km /h (3 mph) and 1/2 hr / 300 m (1000 ft))</ref>

<ref name="legislationgovuk">See definition of "travelling time" in [http://www.legislation.gov.uk/uksi/1996/772/regulation/2/made The Adventure Activities Licensing Regulations 1996, section 2] and [http://www.legislation.gov.uk/uksi/2004/1309/regulation/2/made The Adventure Activities Licensing Regulations 2004, section 2].</ref>

<ref name="MWC">{{cite web |url=http://maumturkswalkingclub.com/mountain-rules/naismiths-rule/ |title=Naismith's rule |publisher=Maumturks Walking Club |access-date=22 January 2017 |archive-date=1 July 2018 |archive-url=https://web.archive.org/web/20180701001533/http://maumturkswalkingclub.com/mountain-rules/naismiths-rule/ |url-status=dead }}</ref>

}}

==External links== * [http://www.sigmadewe.com/fileadmin/user_upload/pdf-Dateien/Bergaufgehen_engl.pdf About walking uphill: time required, energy consumption and the zigzag transition] * [https://web.archive.org/web/20020211065433/http://www.geog.leeds.ac.uk/papers/98-7/ Naismith's Rule] * [https://web.archive.org/web/20150610204527/http://www.mudandroutes.com/archives/24538 Naismith's Rule and Route Timing] * [http://www.mudandroutes.com/archives/24547 Tranter's Correction – is it still relevant?]{{Dead link|date=September 2018 |bot=InternetArchiveBot |fix-attempted=yes }}

Category:Rules of thumb Category:Hiking Category:Navigation Category:1892 introductions