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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Lanchester's_linear_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lanchester's_linear_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Lanchester's linear law</span> </div> </a> <ul id="toc-Lanchester's_linear_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lanchester's_square_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lanchester's_square_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Lanchester's square law</span> </div> </a> <button aria-controls="toc-Lanchester's_square_law-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Lanchester's square law subsection</span> </button> <ul id="toc-Lanchester's_square_law-sublist" class="vector-toc-list"> <li id="toc-Description" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Description"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Description</span> </div> </a> <ul id="toc-Description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Example equations</span> </div> </a> <ul id="toc-Example_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_salvo_combat_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_salvo_combat_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Relation to the salvo combat model</span> </div> </a> <ul id="toc-Relation_to_the_salvo_combat_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lanchester's_law_in_use" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lanchester's_law_in_use"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lanchester's law in use</span> </div> </a> <button aria-controls="toc-Lanchester's_law_in_use-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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Available in 14 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-14" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">14 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%84%D8%A7%D9%86%D9%83%D8%B3%D8%AA%D8%B1" title="قانون لانكستر – Arabic" lang="ar" hreflang="ar" data-title="قانون لانكستر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deddfau_Lanchester" title="Deddfau Lanchester – Welsh" lang="cy" hreflang="cy" data-title="Deddfau Lanchester" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gesetz_von_Lanchester" title="Gesetz von Lanchester – German" lang="de" hreflang="de" data-title="Gesetz von Lanchester" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Leyes_de_Lanchester" title="Leyes de Lanchester – Spanish" lang="es" hreflang="es" data-title="Leyes de Lanchester" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Lois_de_Lanchester" title="Lois de Lanchester – French" lang="fr" hreflang="fr" data-title="Lois de Lanchester" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9E%80%EC%B2%B4%EC%8A%A4%ED%84%B0_%EB%B2%95%EC%B9%99" title="란체스터 법칙 – Korean" lang="ko" hreflang="ko" data-title="란체스터 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7%D7%99_%D7%9C%D7%A0%D7%A6%27%D7%A1%D7%98%D7%A8" title="חוקי לנצ'סטר – Hebrew" lang="he" hreflang="he" data-title="חוקי לנצ'סטר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%83%81%E3%82%A7%E3%82%B9%E3%82%BF%E3%83%BC%E3%81%AE%E6%B3%95%E5%89%87" title="ランチェスターの法則 – Japanese" lang="ja" hreflang="ja" data-title="ランチェスターの法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Leis_de_Lanchester" title="Leis de Lanchester – Portuguese" lang="pt" hreflang="pt" data-title="Leis de Lanchester" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%9E%D1%81%D0%B8%D0%BF%D0%BE%D0%B2%D0%B0_%E2%80%94_%D0%9B%D0%B0%D0%BD%D1%87%D0%B5%D1%81%D1%82%D0%B5%D1%80%D0%B0" title="Законы Осипова — Ланчестера – Russian" lang="ru" hreflang="ru" data-title="Законы Осипова — Ланчестера" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%D8%A7%D8%B3%D8%A7%DA%A9%D8%A7%D9%86%DB%8C_%D9%84%D8%A7%D9%86%DA%86%DB%8E%D8%B3%D8%AA%DB%95%D8%B1" title="یاساکانی لانچێستەر – Central Kurdish" lang="ckb" hreflang="ckb" data-title="یاساکانی لانچێستەر" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lanchesterin_lait" title="Lanchesterin lait – Finnish" lang="fi" hreflang="fi" data-title="Lanchesterin lait" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8_%D0%9B%D0%B0%D0%BD%D1%87%D0%B5%D1%81%D1%82%D0%B5%D1%80%D0%B0" title="Закони Ланчестера – Ukrainian" lang="uk" hreflang="uk" data-title="Закони Ланчестера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%B0%E5%BD%BB%E6%96%AF%E7%89%B9%E6%96%B9%E7%A8%8B" title="兰彻斯特方程 – Chinese" lang="zh" hreflang="zh" data-title="兰彻斯特方程" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Formulae for relative strengths of military forces</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist" style="width:"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="font-size:150%;"><span style="font-size:120%"><a href="/wiki/War" title="War">War</a></span><br /><span class="nobold"><span style="font-size:85%;">(<a href="/wiki/Outline_of_war" title="Outline of war">outline</a>)</span></span></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:War,_northwest_gallery_-_Gari_Melchers,_artist._LCCN91480633_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg/220px-War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg" decoding="async" width="220" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg/330px-War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg/440px-War%2C_northwest_gallery_-_Gari_Melchers%2C_artist._LCCN91480633_%28cropped%29.jpg 2x" data-file-width="3691" data-file-height="1076" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_history" title="Military history">History</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Prehistoric_warfare" title="Prehistoric warfare">Prehistoric</a></li> <li><a href="/wiki/Ancient_warfare" title="Ancient warfare">Ancient</a></li> <li><a href="/wiki/Medieval_warfare" title="Medieval warfare">Post-classical</a> <ul><li><a href="/wiki/Medieval_fortification" title="Medieval fortification">Castles</a></li></ul></li> <li><a href="/wiki/Early_modern_warfare" title="Early modern warfare">Early modern</a> <ul><li><a href="/wiki/Military_Revolution" title="Military Revolution">Military revolution</a></li> <li><a href="/wiki/Pike_and_shot" title="Pike and shot">Pike and shot</a></li> <li><a href="/wiki/Napoleonic_weaponry_and_warfare" title="Napoleonic weaponry and warfare">Napoleonic warfare</a></li></ul></li> <li><a href="/wiki/Modern_warfare" title="Modern warfare">Late modern</a> <ul><li><a href="/wiki/Industrial_warfare" title="Industrial warfare">Industrial warfare</a></li> <li><a href="/wiki/Fourth-generation_warfare" title="Fourth-generation warfare">Fourth-gen warfare</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military" title="Military">Military</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_organization" title="Military organization">Organization</a></li> <li><a href="/wiki/Command_and_control" title="Command and control">Command and control</a></li> <li><a href="/wiki/Defense_ministry" class="mw-redirect" title="Defense ministry">Defense ministry</a></li> <li><a href="/wiki/Army" title="Army">Army</a></li> <li><a href="/wiki/Navy" title="Navy">Navy</a></li> <li><a href="/wiki/Air_force" title="Air force">Air force</a></li> <li><a href="/wiki/Marines" title="Marines">Marines</a></li> <li><a href="/wiki/Coast_guard" title="Coast guard">Coast guard</a></li> <li><a href="/wiki/Space_force" title="Space force">Space force</a></li> <li><a href="/wiki/Military_reserve_force" title="Military reserve force">Reserves</a></li> <li><a href="/wiki/Military_rank" title="Military rank">Ranks</a></li></ul> <hr /> <ul><li><i>Specialties:</i></li> <li><a href="/wiki/Staff_(military)" title="Staff (military)">Staff</a></li> <li><a href="/wiki/Combat_engineer" title="Combat engineer">Engineers</a></li> <li><a href="/wiki/Military_intelligence" title="Military intelligence">Intel</a></li> <li><a href="/wiki/Reconnaissance" title="Reconnaissance">Recon</a></li> <li><a href="/wiki/Medical_corps" title="Medical corps">Medical</a></li> <li><a href="/wiki/Military_police" title="Military police">Police</a></li> <li><a href="/wiki/Military_diving" title="Military diving">Diving</a></li> <li><a href="/wiki/Aircraft_pilot" title="Aircraft pilot">Pilot</a></li></ul> <hr /> <ul><li><i>Land units:</i></li> <li><a href="/wiki/Infantry" title="Infantry">Infantry</a></li> <li><a href="/wiki/Armoured_fighting_vehicle" title="Armoured fighting vehicle">Armor</a> <ul><li><a href="/wiki/Cavalry" title="Cavalry">Cavalry</a></li></ul></li> <li><a href="/wiki/Artillery" title="Artillery">Artillery</a></li> <li><a href="/wiki/Special_forces" title="Special forces">Special forces</a></li> <li><a href="/wiki/Signal_corps" title="Signal corps">Signal corps</a></li></ul> <hr /> <ul><li><i>Naval units:</i></li> <li><a href="/wiki/Warship" title="Warship">Warships</a></li> <li><a href="/wiki/Submarine" title="Submarine">Submarines</a></li> <li><a href="/wiki/Aircraft_carrier" title="Aircraft carrier">Aircraft carriers</a></li> <li><a href="/wiki/Landing_craft" title="Landing craft">Landing craft</a></li> <li><a href="/wiki/Auxiliary_ship" title="Auxiliary ship">Auxiliary ships</a></li></ul> <hr /> <ul><li><i>Air units:</i></li> <li><a href="/wiki/Fighter_aircraft" title="Fighter aircraft">Fighters</a></li> <li><a href="/wiki/Bomber" title="Bomber">Bombers</a></li> <li><a href="/wiki/Airborne_early_warning_and_control" title="Airborne early warning and control">Command</a></li> <li><a href="/wiki/Close_air_support" title="Close air support">Close air support</a></li> <li><a href="/wiki/Electronic-warfare_aircraft" title="Electronic-warfare aircraft">Electronic-warfare</a></li> <li><a href="/wiki/Reconnaissance_aircraft" title="Reconnaissance aircraft">Reconnaissance</a></li></ul> <hr /> <ul><li><i>Combat systems:</i></li> <li><a href="/wiki/Fire-control_system" title="Fire-control system">Fire-control system</a></li> <li><a href="/wiki/Fire-control_radar" title="Fire-control radar">Fire-control radar</a></li> <li><a href="/wiki/Director_(military)" title="Director (military)">Director (military)</a></li> <li><a href="/wiki/Combat_information_center" title="Combat information center">Combat information center</a></li> <li><a href="/wiki/Sonar" title="Sonar">Sonar</a></li> <li><a href="/wiki/Radar" title="Radar">Radar</a></li></ul> <hr /> <ul><li><i>Historical:</i></li> <li><a href="/wiki/Ship_gun_fire-control_system" title="Ship gun fire-control system">Ship gun fire-control</a></li> <li><a href="/wiki/Gun_data_computer" title="Gun data computer">Gun data computer</a></li> <li><a href="/wiki/Torpedo_Data_Computer" title="Torpedo Data Computer">Torpedo data computer</a></li></ul> <hr /> <ul><li><i>Development:</i></li> <li><a href="/wiki/Basic_training" class="mw-redirect" title="Basic training">Basic training</a></li> <li><a href="/wiki/Military_exercise" title="Military exercise">Military manoeuvrers</a></li> <li><a href="/wiki/Combat_training" class="mw-redirect" title="Combat training">Combat training</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Battlespace" title="Battlespace">Battlespace</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Aerospace" title="Aerospace">Aerospace</a> <ul><li><a href="/wiki/Aerial_warfare" title="Aerial warfare">Aerial</a></li> <li><a href="/wiki/Airborne_forces" title="Airborne forces">Airborne</a></li> <li><a href="/wiki/Space_warfare" title="Space warfare">Space</a></li></ul></li></ul> <hr /> <ul><li><a href="/wiki/Land_warfare" title="Land warfare">Land</a> <ul><li><a href="/wiki/Cold-weather_warfare" title="Cold-weather warfare">Cold-region</a></li> <li><a href="/wiki/Desert_warfare" title="Desert warfare">Desert</a></li> <li><a href="/wiki/Fortification" title="Fortification">Fortification</a></li> <li><a href="/wiki/Jungle_warfare" title="Jungle warfare">Jungle</a></li> <li><a href="/wiki/Mountain_warfare" title="Mountain warfare">Mountain</a></li> <li><a href="/wiki/Urban_warfare" title="Urban warfare">Urban</a></li> <li><a href="/wiki/Subterranean_warfare" title="Subterranean warfare">Subterranean</a> <ul><li><a href="/wiki/Tunnel_warfare" title="Tunnel warfare">Tunnel</a></li></ul></li></ul></li></ul> <hr /> <ul><li><a href="/wiki/Naval_warfare" title="Naval warfare">Sea</a> <ul><li><a href="/wiki/Amphibious_warfare" title="Amphibious warfare">Amphibious</a></li> <li><a href="/wiki/Blue-water_navy" title="Blue-water navy">Blue</a></li> <li><a href="/wiki/Brown-water_navy" title="Brown-water navy">Brown</a></li> <li><a href="/wiki/Green-water_navy" title="Green-water navy">Green</a></li> <li><a href="/wiki/Surface_warfare" title="Surface warfare">Surface</a></li> <li><a href="/wiki/Underwater_warfare" title="Underwater warfare">Underwater</a> <ul><li><a href="/wiki/Seabed_warfare" title="Seabed warfare">Seabed</a></li></ul></li></ul></li></ul> <hr /> <ul><li><a href="/wiki/Cyberwarfare" title="Cyberwarfare">Cyber</a></li> <li><a href="/wiki/Information_warfare" title="Information warfare">Information</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Weapon" title="Weapon">Weapons</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anti-aircraft_warfare" title="Anti-aircraft warfare">Air defence</a></li> <li><a href="/wiki/Armoured_warfare" title="Armoured warfare">Armor</a></li> <li><a href="/wiki/Artillery" title="Artillery">Artillery</a></li> <li><a href="/wiki/Barrage_(artillery)" title="Barrage (artillery)">Barrage</a></li> <li><a href="/wiki/Biological_warfare" title="Biological warfare">Biological</a></li> <li><a href="/wiki/Military_camouflage" title="Military camouflage">Camouflage</a></li> <li><a href="/wiki/Cavalry" title="Cavalry">Cavalry</a> <ul><li><a href="/wiki/Horses_in_warfare" title="Horses in warfare">Horses</a></li> <li><a href="/wiki/Cavalry_tactics#Cavalry_in_modern_warfare" title="Cavalry tactics">Air cavalry</a></li></ul></li> <li><a href="/wiki/Chemical_warfare" title="Chemical warfare">Chemical</a></li> <li><a href="/wiki/Combined_arms" title="Combined arms">Combined arms</a></li> <li><a href="/wiki/Conventional_warfare" title="Conventional warfare">Conventional</a></li> <li><a href="/wiki/Cyberweapon" title="Cyberweapon">Cyber</a></li> <li><a href="/wiki/Area_denial_weapon" title="Area denial weapon">Denial</a></li> <li><a href="/wiki/Disinformation" title="Disinformation">Disinformation</a></li> <li><a href="/wiki/Drone_warfare" title="Drone warfare">Drone</a> / <a href="/wiki/Military_robot" title="Military robot">Robot</a></li> <li><a href="/wiki/Electromagnetic_warfare" class="mw-redirect" title="Electromagnetic warfare">Electromagnetic</a></li> <li><a href="/wiki/Infantry" title="Infantry">Infantry</a></li> <li><a href="/wiki/Loitering_munition" title="Loitering munition">Loitering</a></li> <li><a href="/wiki/Missile_defense" title="Missile defense">Missile</a></li> <li><a href="/wiki/Martial_music" title="Martial music">Music</a></li> <li><a href="/wiki/Nuclear_warfare" title="Nuclear warfare">Nuclear</a></li> <li><a href="/wiki/Psychological_warfare" title="Psychological warfare">Psychological</a></li> <li><a href="/wiki/Radiological_warfare" title="Radiological warfare">Radiological</a></li> <li><a href="/wiki/Submarine_warfare" title="Submarine warfare">Submarine</a></li> <li><a href="/wiki/Unconventional_warfare" title="Unconventional warfare">Unconventional</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_tactics" title="Military tactics">Tactics</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/List_of_military_tactics" title="List of military tactics">List of military tactics</a> <ul><li><a href="/wiki/Air_combat_manoeuvring" title="Air combat manoeuvring">Aerial</a></li> <li><a href="/wiki/Airlift" title="Airlift">Airlift</a> <ul><li><a href="/wiki/Air_assault" title="Air assault">Air assault</a></li> <li><a href="/wiki/Airbridge_(logistics)" title="Airbridge (logistics)">Airbridge</a></li> <li><a href="/wiki/Airdrop" title="Airdrop">Airdrop</a></li></ul></li> <li><a href="/wiki/Anti-aircraft_warfare" title="Anti-aircraft warfare">Anti-aircraft</a></li> <li><a href="/wiki/Anti-submarine_warfare" title="Anti-submarine warfare">Anti-sub</a></li> <li><a href="/wiki/Anti-tank_warfare" title="Anti-tank warfare">Anti-tank</a></li> <li><a href="/wiki/Battle" title="Battle">Battle</a></li> <li><a href="/wiki/Cavalry_tactics" title="Cavalry tactics">Cavalry</a></li> <li><a href="/wiki/Charge_(warfare)" title="Charge (warfare)">Charge</a></li> <li><a href="/wiki/Close-quarters_battle" title="Close-quarters battle">CQC</a></li> <li><a href="/wiki/Counterattack" title="Counterattack">Counterattack</a></li> <li><a href="/wiki/Counterinsurgency" title="Counterinsurgency">Counterinsurgency</a></li> <li><a href="/wiki/Convoy" title="Convoy">Convoy</a></li> <li><a href="/wiki/Defeat_in_detail" title="Defeat in detail">Defeat in detail</a></li> <li><a href="/wiki/Defensive_fighting_position" title="Defensive fighting position">Foxhole</a></li> <li><a href="/wiki/Drone_warfare" title="Drone warfare">Drone</a></li> <li><a href="/wiki/Envelopment" title="Envelopment">Envelopment</a></li> <li><a href="/wiki/Tactical_formation" title="Tactical formation">Formation</a></li> <li><a href="/wiki/Guerrilla_warfare" title="Guerrilla warfare">Guerrilla</a></li> <li><a href="/wiki/Naval_tactics" title="Naval tactics">Naval</a></li> <li><a href="/wiki/Shock_and_awe" title="Shock and awe">Rapid dominance</a></li> <li><a href="/wiki/Encirclement" title="Encirclement">Encirclement</a></li> <li><a href="/wiki/Investment_(military)" title="Investment (military)">Investment</a> <ul><li><a href="/wiki/Siege" title="Siege">Siege</a></li></ul></li> <li><a href="/wiki/Swarming_(military)" title="Swarming (military)">Swarm</a></li> <li><a href="/wiki/Screening_(tactical)" title="Screening (tactical)">Screen</a></li> <li><a href="/wiki/Tactical_objective" title="Tactical objective">Tactical objective</a></li> <li><a href="/wiki/Saturation_attack" title="Saturation attack">Target saturation</a></li> <li><a href="/wiki/Trench_warfare" title="Trench warfare">Trench</a></li> <li><a href="/wiki/Withdrawal_(military)" title="Withdrawal (military)">Withdrawal</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Operational_level_of_war" title="Operational level of war">Operational</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_operation" title="Military operation">Military operation</a> <ul><li><a href="/wiki/Special_operations" title="Special operations">Special</a></li></ul></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><i><a href="/wiki/Blitzkrieg" title="Blitzkrieg">Blitzkrieg</a></i></li> <li><a href="/wiki/Expeditionary_warfare" title="Expeditionary warfare">Expeditionary</a></li> <li><a href="/wiki/Deep_operation" title="Deep operation">Deep operation</a></li> <li><a href="/wiki/Maneuver_warfare" title="Maneuver warfare">Maneuver</a></li> <li><a href="/wiki/Operational_manoeuvre_group" title="Operational manoeuvre group">Operational manoeuvre group</a></li> <li><a href="/wiki/Raid_(military)" title="Raid (military)">Raid</a></li> <li><a href="/wiki/Covert_operation" title="Covert operation">Covert</a></li> <li><a href="/wiki/Stay-behind" title="Stay-behind">Stay-behind</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_strategy" title="Military strategy">Strategy</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/List_of_military_strategies_and_concepts" title="List of military strategies and concepts">List of military strategies and concepts</a> <ul><li><a href="/wiki/Military_campaign" title="Military campaign">Military campaign</a></li> <li><a href="/wiki/Anti-access/area_denial" title="Anti-access/area denial">Anti-access</a></li> <li><a href="/wiki/Attrition_warfare" title="Attrition warfare">Attrition</a></li> <li><a href="/wiki/Commerce_raiding" title="Commerce raiding">Commerce raiding</a></li> <li><a href="/wiki/Counter-offensive" class="mw-redirect" title="Counter-offensive">Counter-offensive</a></li> <li><a href="/wiki/Culminating_point" title="Culminating point">Culminating</a></li> <li><a href="/wiki/Defence_in_depth" title="Defence in depth">Defence in depth</a></li> <li><a href="/wiki/Fabian_strategy" title="Fabian strategy">Fabian</a></li> <li><a href="/wiki/Empty_Fort_Strategy" title="Empty Fort Strategy">Empty fort</a></li> <li><a href="/wiki/Mosaic_Warfare" class="mw-redirect" title="Mosaic Warfare">Mosaic</a></li> <li><a href="/wiki/Military_deception" title="Military deception">Deception</a></li> <li><a href="/wiki/Strategic_defence" title="Strategic defence">Defensive</a></li> <li><a href="/wiki/Strategic_depth" title="Strategic depth">Depth</a></li> <li><a href="/wiki/Strategic_goal_(military)" title="Strategic goal (military)">Goal</a></li> <li><a href="/wiki/Nuclear_strategy" title="Nuclear strategy">Nuclear</a></li> <li><a href="/wiki/Naval_strategy" title="Naval strategy">Naval</a></li> <li><a href="/wiki/Offensive_(military)" title="Offensive (military)">Offensive</a></li> <li><a href="/wiki/Scorched_earth" title="Scorched earth">Scorched earth</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Grand_strategy" title="Grand strategy">Grand strategy</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Asymmetric_warfare" title="Asymmetric warfare">Asymmetric</a></li> <li><a href="/wiki/Blockade" title="Blockade">Blockade</a></li> <li><a href="/wiki/Broken-backed_war_theory" title="Broken-backed war theory">Broken-backed</a></li> <li><a href="/wiki/Class_conflict" title="Class conflict">Class</a></li> <li><a href="/wiki/Cold_war_(general_term)" class="mw-redirect" title="Cold war (general term)">Cold war</a></li> <li><a href="/wiki/Colonial_war" title="Colonial war">Colonial</a></li> <li><a href="/wiki/Conquest" title="Conquest">Conquest</a></li> <li><a href="/wiki/Containment" title="Containment">Containment</a></li> <li><a href="/wiki/Divide_and_rule" title="Divide and rule">Divide and conquer</a></li> <li><a href="/wiki/Economic_warfare" title="Economic warfare">Economic</a></li> <li><a href="/wiki/Endemic_warfare" class="mw-redirect" title="Endemic warfare">Endemic</a></li> <li><a href="/wiki/Fleet_in_being" title="Fleet in being">Fleet in being</a></li> <li><a href="/wiki/Irregular_warfare" title="Irregular warfare">Irregular</a></li> <li><a href="/wiki/Wars_of_national_liberation" title="Wars of national liberation">Liberation</a></li> <li><a href="/wiki/Limited_war" title="Limited war">Limited</a></li> <li><a href="/wiki/Network-centric_warfare" title="Network-centric warfare">Network-centric</a></li> <li><a href="/wiki/New_generation_warfare" title="New generation warfare">New generation</a></li> <li><a href="/wiki/Perpetual_war" title="Perpetual war">Perpetual</a></li> <li><a href="/wiki/Political_warfare" title="Political warfare">Political</a></li> <li><a href="/wiki/Princely_rebellion" title="Princely rebellion">Princely</a></li> <li><a href="/wiki/Proxy_war" title="Proxy war">Proxy</a></li> <li><a href="/wiki/Religious_war" title="Religious war">Religious</a></li> <li><a href="/wiki/Resource_war" title="Resource war">Resource</a></li> <li><a href="/wiki/Strategic_studies" title="Strategic studies">Strategic</a></li> <li><a href="/wiki/War_of_succession" title="War of succession">Succession</a></li> <li><a href="/wiki/Military_technology" title="Military technology">Technology</a></li> <li><a href="/wiki/Theater_(warfare)" title="Theater (warfare)">Theater</a></li> <li><a href="/wiki/Total_war" title="Total war">Total war</a></li> <li><a href="/wiki/World_war" title="World war">World war</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_administration" title="Military administration">Administrative</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_branch" title="Military branch">Branch</a></li> <li><a href="/wiki/Military_policy" title="Military policy">Policy</a></li> <li><a href="/wiki/Staff_(military)" title="Staff (military)">Staff</a></li> <li><a href="/wiki/Military_education_and_training" title="Military education and training">Training</a></li> <li><a href="/wiki/Military_service" title="Military service">Service</a></li> <li><a href="/wiki/Military_sociology" title="Military sociology">Sociology</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_organization" title="Military organization">Organization</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Area_of_responsibility" title="Area of responsibility">Area of responsibility</a></li> <li><a href="/wiki/Command_hierarchy" title="Command hierarchy">Chain of command</a></li> <li><a href="/wiki/Command_and_control" title="Command and control">Command and control</a></li> <li><a href="/wiki/Military_doctrine" title="Military doctrine">Doctrine</a></li> <li><a href="/wiki/Order_of_battle" title="Order of battle">Order of battle</a></li> <li><a href="/wiki/Principles_of_war" title="Principles of war">Principles of war</a></li> <li><a href="/wiki/Economy_of_force" title="Economy of force">Economy of force</a></li> <li><a href="/wiki/Military_medicine" title="Military medicine">Medicine</a></li> <li><a href="/wiki/Military_engineering" title="Military engineering">Engineers</a></li> <li><a href="/wiki/Military_intelligence" title="Military intelligence">Intelligence</a></li> <li><a href="/wiki/Military_rank" title="Military rank">Ranks</a></li> <li><a href="/wiki/Military_technology" title="Military technology">Technology and equipment</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_personnel" title="Military personnel">Personnel</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_recruitment" title="Military recruitment">Recruitment</a></li> <li><a href="/wiki/Conscription" title="Conscription">Conscription</a></li> <li><a href="/wiki/Draft_evasion" title="Draft evasion">Draft evasion</a></li> <li><a href="/wiki/Mobilization" title="Mobilization">Mobilization</a></li> <li><a href="/wiki/Recruit_training" class="mw-redirect" title="Recruit training">Training</a></li> <li><a href="/wiki/Military_specialism" title="Military specialism">Specialism</a></li> <li><a href="/wiki/Soldier" title="Soldier">Soldier</a></li> <li><a href="/wiki/Morale" title="Morale">Morale</a></li> <li><a href="/wiki/Military_volunteer" title="Military volunteer">Volunteer</a></li> <li><a href="/wiki/Women_in_the_military" title="Women in the military">Women</a></li> <li><a href="/wiki/Children_in_the_military" title="Children in the military">Children</a></li> <li><a href="/wiki/Transgender_people_and_military_service" title="Transgender people and military service">Transgender</a></li> <li><a href="/wiki/Sexual_harassment_in_the_military" title="Sexual harassment in the military">Harassment</a></li> <li><a href="/wiki/Conscientious_objector" title="Conscientious objector">Conscientious objector</a></li> <li><a href="/wiki/Conscription_and_sexism" title="Conscription and sexism">Conscription and sexism</a></li> <li><a href="/wiki/Counter-recruitment" title="Counter-recruitment">Counter-recruitment</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_logistics" title="Military logistics">Logistics</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_military_logistics" title="History of military logistics">History</a></li> <li><a href="/wiki/War_economy" title="War economy">War economy</a></li> <li><a href="/wiki/Arms_industry" title="Arms industry">Arms industry</a></li> <li><a href="/wiki/Materiel" title="Materiel">Materiel</a></li> <li><a href="/wiki/Military_supply-chain_management" title="Military supply-chain management">Supply-chain management</a></li> <li><a href="/wiki/Military%E2%80%93industrial_complex" title="Military–industrial complex">Military–industrial complex</a></li> <li><a href="/wiki/Military_base" title="Military base">Base</a> <ul><li><a href="/wiki/Main_operating_base" title="Main operating base">MOB</a></li> <li><a href="/wiki/Forward_operating_base" title="Forward operating base">FOB</a></li> <li><a href="/wiki/Outpost_(military)" title="Outpost (military)">Outpost</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_science" title="Military science">Science</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Power_projection" title="Power projection">Power projection</a></li> <li><a href="/wiki/Loss-of-strength_gradient" title="Loss-of-strength gradient">Loss-of-strength gradient</a></li> <li><a class="mw-selflink selflink">Lanchester's laws</a> <ul><li><a href="/wiki/Force_multiplication" title="Force multiplication">Force multiplication</a></li></ul></li> <li><a href="/wiki/Morale" title="Morale">Morale</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Law_of_war" title="Law of war">Law</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_occupation" title="Military occupation">Occupation</a></li> <li><a href="/wiki/Armistice" title="Armistice">Armistice</a></li> <li><a href="/wiki/Ceasefire" title="Ceasefire">Ceasefire</a></li> <li><a href="/wiki/Court-martial" title="Court-martial">Court-martial</a></li> <li><a href="/wiki/Desertion" title="Desertion">Desertion</a></li> <li><a href="/wiki/Geneva_Conventions" title="Geneva Conventions">Geneva Conventions</a></li> <li><a href="/wiki/Geneva_Protocol" title="Geneva Protocol">Geneva Protocol</a></li> <li><a href="/wiki/Islamic_military_jurisprudence" title="Islamic military jurisprudence">Islamic rules</a></li> <li><a href="/wiki/Military_justice" title="Military justice">Justice</a></li> <li><a href="/wiki/Combatant" title="Combatant">Lawful</a> / <a href="/wiki/Unlawful_combatant" title="Unlawful combatant">Unlawful combatant</a></li> <li><a href="/wiki/Perfidy" title="Perfidy">Perfidy</a></li> <li><a href="/wiki/Regular_army" title="Regular army">Regular</a> / <a href="/wiki/Irregular_military" title="Irregular military">Irregular</a></li> <li><a href="/wiki/Jewish_laws_on_war" class="mw-redirect" title="Jewish laws on war">Jewish laws on war</a></li> <li><a href="/wiki/Right_of_conquest" title="Right of conquest">Right of conquest</a></li> <li><a href="/wiki/Self-defence_in_international_law" title="Self-defence in international law">Right of self-defence</a></li> <li><a href="/wiki/Rules_of_engagement" title="Rules of engagement">Rules of engagement</a></li> <li><a href="/wiki/Self-determination" title="Self-determination">Self-determination</a></li> <li><a href="/wiki/Martial_law" title="Martial law">Martial law</a></li> <li><a href="/wiki/War_crime" title="War crime">War crime</a></li> <li><a href="/wiki/War_treason" title="War treason">War treason</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_theory" title="Military theory">Theory</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Air_supremacy" title="Air supremacy">Air supremacy</a></li> <li><a href="/wiki/Appeasement" title="Appeasement">Appeasement</a></li> <li><a href="/wiki/Command_of_the_sea" title="Command of the sea">Command of the sea</a></li> <li><a href="/wiki/Deterrence_theory" title="Deterrence theory">Deterrence theory</a></li> <li><a href="/wiki/Full-spectrum_dominance" title="Full-spectrum dominance">Full-spectrum dominance</a></li> <li><a href="/wiki/Overmatch" title="Overmatch">Overmatch</a></li> <li><i><a href="/w/index.php?title=Unrestricted_warfare&action=edit&redlink=1" class="new" title="Unrestricted warfare (page does not exist)">Unrestricted warfare</a></i></li> <li><a href="/wiki/Just_war_theory" title="Just war theory">Just war theory</a></li> <li><a href="/wiki/Principles_of_war" title="Principles of war">Principles of war</a></li> <li><a href="/wiki/Philosophy_of_war" title="Philosophy of war">Philosophy of war</a></li> <li><a href="/wiki/Security_dilemma" title="Security dilemma">Security dilemma</a> <ul><li><a href="/wiki/Tripwire_force" title="Tripwire force">Tripwire force</a></li></ul></li> <li><a href="/wiki/Professional_wargaming" title="Professional wargaming">Wargaming</a> <ul><li><a href="/wiki/Military_simulation" title="Military simulation">Simulation</a></li> <li><a href="/wiki/Military_exercise" title="Military exercise">Exercises</a></li></ul></li> <li><a href="/wiki/Combat_effectiveness" title="Combat effectiveness">Combat effectiveness</a></li> <li><a href="/wiki/Center_of_gravity_(military)" title="Center of gravity (military)">Center of gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_operations_other_than_war" title="Military operations other than war">Non-warfare</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arms_control" title="Arms control">Arms control</a></li> <li><a href="/wiki/Counter-insurgency" class="mw-redirect" title="Counter-insurgency">Counter-insurgency</a></li> <li><a href="/wiki/Deterrence_theory" title="Deterrence theory">Deterrence</a></li> <li><a href="/wiki/Disaster_response" title="Disaster response">Disaster response</a></li> <li><a href="/wiki/Grey-zone_(international_relations)" title="Grey-zone (international relations)">Grey-zone</a></li> <li><a href="/wiki/Gunboat_diplomacy" title="Gunboat diplomacy">Gunboat diplomacy</a></li> <li><a href="/wiki/Humanitarian_aid" title="Humanitarian aid">Humanitarian aid</a></li> <li><a href="/wiki/Law_enforcement" title="Law enforcement">Law enforcement</a></li> <li><a href="/wiki/Low-intensity_conflict" title="Low-intensity conflict">Low-intensity conflict</a></li> <li><a href="/wiki/Military_engineering" title="Military engineering">Military engineering</a></li> <li><a href="/wiki/Peacekeeping" title="Peacekeeping">Peacekeeping</a> <ul><li><a href="/wiki/Peacebuilding" title="Peacebuilding">Peacebuilding</a></li> <li><a href="/wiki/Peace_through_strength" title="Peace through strength">Peace through strength</a></li></ul></li> <li><a href="/wiki/Show_of_force" title="Show of force">Show of force</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Military_tradition" title="Military tradition">Culture</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Military_awards_and_decorations" title="Military awards and decorations">Awards and decorations</a></li> <li><a href="/wiki/Battle_cry" title="Battle cry">Battle cry</a></li> <li><a href="/wiki/Warrior" title="Warrior">Warrior</a></li> <li><a href="/wiki/War_film" title="War film">War film</a></li> <li><a href="/wiki/Military_science_fiction" title="Military science fiction">Military science fiction</a></li> <li><a href="/wiki/War_novel" title="War novel">War novel</a></li> <li><a href="/wiki/Anti-war_movement" title="Anti-war movement">Anti-war movement</a></li> <li><a href="/wiki/Foot_drill" title="Foot drill">Foot drill</a></li> <li><a href="/wiki/War_song" title="War song">War song</a></li> <li><a href="/wiki/Military_uniform" title="Military uniform">Uniform</a></li> <li><a href="/wiki/Wargame" title="Wargame">Wargame</a></li> <li><a href="/wiki/Militarization" title="Militarization">Militarization</a> / <a href="/wiki/Militarism" title="Militarism">Militarism</a> <ul><li><a href="/wiki/Antimilitarism" title="Antimilitarism">Antimilitarism</a></li></ul></li> <li><a href="/wiki/Camp_follower" title="Camp follower">Camp follower</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)">Books</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Seven_Military_Classics" title="Seven Military Classics">Seven Military Classics</a> <ul><li><a href="/wiki/The_Art_of_War" title="The Art of War">The Art of War</a></li></ul></li> <li><a href="/wiki/De_re_militari" title="De re militari">De re militari</a></li> <li><a href="/wiki/Strategemata" title="Strategemata">Strategemata</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)">Related</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Women_in_war" title="Women in war">Women in war</a></li> <li><a href="/wiki/War_resister" title="War resister">War resister</a></li> <li><a href="/wiki/War_studies" title="War studies">War studies</a></li> <li><a href="/wiki/Horses_in_warfare" title="Horses in warfare">Horses in warfare</a></li> <li><a href="/wiki/Wartime_sexual_violence" title="Wartime sexual violence">Wartime sexual violence</a></li> <li><a href="/wiki/Fifth_column" title="Fifth column">Fifth column</a></li> <li><a href="/wiki/Mercenary" title="Mercenary">Mercenary</a></li> <li><a href="/wiki/Pacifism" title="Pacifism">Pacifism</a></li> <li><a href="/wiki/Privateer" title="Privateer">Privateer</a></li> <li><a href="/wiki/Private_military_company" title="Private military company">Private military company</a></li> <li><a href="/wiki/Soldiers_are_murderers" title="Soldiers are murderers">Soldiers are murderers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;;color: var(--color-base)"><a href="/wiki/Category:Military_lists" title="Category:Military lists">Lists</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Lists_of_battles" title="Lists of battles">Battles</a></li> <li><a href="/wiki/List_of_military_occupations" title="List of military occupations">Military occupations</a></li> <li><a href="/wiki/List_of_established_military_terms" title="List of established military terms">Military terms</a></li> <li><a href="/wiki/List_of_military_operations" title="List of military operations">Operations</a></li> <li><a href="/wiki/List_of_sieges" title="List of sieges">Sieges</a></li> <li><a href="/wiki/List_of_war_crimes" title="List of war crimes">War crimes</a></li> <li><a href="/wiki/Outline_of_war#Wars" title="Outline of war">Wars</a></li> <li><a href="/wiki/Lists_of_weapons" title="Lists of weapons">Weapons</a></li> <li><a href="/wiki/List_of_military_writers" title="List of military writers">Writers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:War" title="Template:War"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:War" title="Template talk:War"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:War" title="Special:EditPage/Template:War"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Lanchester's laws</b> are mathematical <a href="/wiki/Formula" title="Formula">formulas</a> for calculating the relative strengths of <a href="/wiki/Military_force" class="mw-redirect" title="Military force">military forces</a>. The Lanchester equations are <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> describing the time dependence of two armies' strengths A and B as a function of time, with the function depending only on A and B.<sup id="cite_ref-lancs_1-0" class="reference"><a href="#cite_note-lancs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-davis95_2-0" class="reference"><a href="#cite_note-davis95-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In 1915 and 1916 during <a href="/wiki/World_War_I" title="World War I">World War I</a>, M. Osipov<sup id="cite_ref-osipov_3-0" class="reference"><a href="#cite_note-osipov-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: vii–viii">: vii–viii </span></sup> and <a href="/wiki/Frederick_Lanchester" class="mw-redirect" title="Frederick Lanchester">Frederick Lanchester</a> independently devised a series of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> to demonstrate the power relationships between opposing forces.<sup id="cite_ref-wrigge95_4-0" class="reference"><a href="#cite_note-wrigge95-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Among these are what is known as <i>Lanchester's linear law</i> (for <a href="/wiki/Ancient_warfare" title="Ancient warfare">ancient combat</a>) and <i>Lanchester's square law</i> (for <a href="/wiki/Modern_warfare" title="Modern warfare">modern combat</a> with long-range weapons such as firearms). </p><p>As of 2017 modified variations of the Lanchester equations continue to form the basis of analysis in many of the US Army’s combat simulations,<sup id="cite_ref-christian19_5-0" class="reference"><a href="#cite_note-christian19-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and in 2016 a RAND Corporation report examined by these laws the probable outcome in the event of a Russian invasion into the Baltic nations of Estonia, Latvia, and Lithuania.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Lanchester's_linear_law"><span id="Lanchester.27s_linear_law"></span>Lanchester's linear law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=1" title="Edit section: Lanchester's linear law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For ancient combat, between <a href="/wiki/Phalanx_formation" class="mw-redirect" title="Phalanx formation">phalanxes</a> of soldiers with <a href="/wiki/Spear" title="Spear">spears</a> for example, one soldier could only ever fight exactly one other soldier at a time. If each soldier kills, and is killed by, exactly one other, then the number of soldiers remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons. </p><p>The linear law also applies to unaimed fire into an enemy-occupied area. The rate of attrition depends on the density of the available targets in the target area as well as the number of weapons shooting. If two forces, occupying the same land area and using the same weapons, shoot randomly into the same target area, they will both suffer the same rate and number of casualties, until the smaller force is eventually eliminated: the greater probability of any one shot hitting the larger force is balanced by the greater number of shots directed at the smaller force. </p> <div class="mw-heading mw-heading2"><h2 id="Lanchester's_square_law"><span id="Lanchester.27s_square_law"></span>Lanchester's square law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=2" title="Edit section: Lanchester's square law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Lanchester's square law</b> is also known as the <b>N-square law</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Description">Description</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=3" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Damagerace.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Damagerace.JPG/220px-Damagerace.JPG" decoding="async" width="220" height="586" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Damagerace.JPG/330px-Damagerace.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/9/92/Damagerace.JPG 2x" data-file-width="346" data-file-height="921" /></a><figcaption>Idealized simulation of two forces damaging each other neglecting all other circumstances than the 1) Size of army 2) Rate of damaging. The picture illustrates the principle of Lanchester's square law.</figcaption></figure> <p>With firearms engaging each other directly with aimed shooting from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons shooting. Lanchester determined that the power of such a force is proportional not to the number of <a href="/wiki/Military_unit" class="mw-redirect" title="Military unit">units</a> it has, but to the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of the number of units. This is known as Lanchester's square law. </p><p>More precisely, the law specifies the casualties a shooting force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition. It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each unit (soldier, ship, etc.) can kill only one equivalent unit at a time. For this reason, the law does not apply to machine guns, artillery with unguided munitions, or nuclear weapons. The law requires an assumption that casualties accumulate over time: it does not work in situations in which opposing troops kill each other instantly, either by shooting simultaneously or by one side getting off the first shot and inflicting multiple casualties. </p><p>Note that Lanchester's square law does not apply to technological force, only numerical force; so it requires an N-squared-fold increase in quality to compensate for an N-fold decrease in quantity. </p> <div class="mw-heading mw-heading3"><h3 id="Example_equations">Example equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=4" title="Edit section: Example equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that two armies, Red and Blue, are engaging each other in combat. Red is shooting a continuous stream of bullets at Blue. Meanwhile, Blue is shooting a continuous stream of bullets at Red. </p><p>Let symbol <i>A</i> represent the number of soldiers in the Red force. Each one has <i>offensive firepower α</i>, which is the number of enemy soldiers it can incapacitate (e.g., kill or injure) per unit time. Likewise, Blue has <i>B</i> soldiers, each with offensive firepower <i>β</i>. </p><p>Lanchester's square law calculates the number of soldiers lost on each side using the following pair of equations.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Here, <i>dA/dt</i> represents the rate at which the number of Red soldiers is changing at a particular instant. A negative value indicates the loss of soldiers. Similarly, <i>dB/dt</i> represents the rate of change of the number of Blue soldiers. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}=-\beta B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>A</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}=-\beta B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385be9df1f943b092a6a07bee7757149199198dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}=-\beta B}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} B}{\mathrm {d} t}}=-\alpha A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>B</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} B}{\mathrm {d} t}}=-\alpha A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a2a89e5bea54f41680d4f6fd0ed9bb5beaebf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.03ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} B}{\mathrm {d} t}}=-\alpha A}" /></span></dd></dl> <p>The solution to these equations shows that: </p> <ul><li>If <i>α</i>=<i>β</i>, i.e. the two sides have equal firepower, the side with more soldiers at the beginning of the battle will win;</li> <li>If <i>A</i>=<i>B</i>, i.e. the two sides have equal numbers of soldiers, the side with greater firepower will win;</li> <li>If <i>A</i>><i>B</i> and <i>α</i>><i>β</i>, then Red will win, while if <i>A</i><<i>B</i> and <i>α</i><<i>β</i>, Blue will win;</li> <li>If <i>A</i>><i>B</i> but <i>α</i><<i>β</i>, or <i>A</i><<i>B</i> but <i>α</i>><i>β</i>, the winning side will depend on whether the ratio of <i>β</i>/<i>α</i> is greater or less than the square of the ratio of <i>A</i>/<i>B</i>. Thus, if numbers and firepower are unequal in opposite directions, a superiority in firepower equal to the square of the inferiority in numbers is required for victory; or, to put it another way, the effectiveness of the army rises proportionate to the square of the number of people in it, but only linearly with their fighting ability.</li></ul> <p>The first three of these conclusions are obvious. The final one is the origin of the name "square law". </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_salvo_combat_model">Relation to the salvo combat model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=5" title="Edit section: Relation to the salvo combat model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lanchester's equations are related to the more recent <a href="/wiki/Salvo_combat_model" title="Salvo combat model">salvo combat model</a> equations, with two main differences. </p><p>First, Lanchester's original equations form a continuous time model, whereas the basic salvo equations form a discrete time model. In a gun battle, bullets or shells are typically fired in large quantities. Each round has a relatively low chance of hitting its target, and does a relatively small amount of damage. Therefore, Lanchester's equations model gunfire as a stream of firepower that continuously weakens the enemy force over time. </p><p>By comparison, cruise missiles typically are fired in relatively small quantities. Each one has a high probability of hitting its target, and carries a relatively powerful warhead. Therefore, it makes more sense to model them as a discrete pulse (or salvo) of firepower in a discrete time model. </p><p>Second, Lanchester's equations include only offensive firepower, whereas the salvo equations also include defensive firepower. Given their small size and large number, it is not practical to intercept bullets and shells in a gun battle. By comparison, cruise missiles can be intercepted (shot down) by surface-to-air missiles and anti-aircraft guns. Therefore, missile combat models include those active defenses. </p> <div class="mw-heading mw-heading2"><h2 id="Lanchester's_law_in_use"><span id="Lanchester.27s_law_in_use"></span>Lanchester's law in use</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=6" title="Edit section: Lanchester's law in use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lanchester's laws have been used to model historical battles for research purposes. Examples include <a href="/wiki/Pickett%27s_Charge" title="Pickett's Charge">Pickett's Charge</a> of Confederate infantry against Union infantry during the 1863 <a href="/wiki/Battle_of_Gettysburg" title="Battle of Gettysburg">Battle of Gettysburg</a>,<sup id="cite_ref-armstrong15_8-0" class="reference"><a href="#cite_note-armstrong15-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> the 1940 <a href="/wiki/Battle_of_Britain" title="Battle of Britain">Battle of Britain</a> between the British and German air forces,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Battle_of_Kursk" title="Battle of Kursk">Battle of Kursk</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In modern warfare, to take into account that to some extent both linear and the square apply often, an exponent of 1.5 is used.<sup id="cite_ref-simpkin_11-0" class="reference"><a href="#cite_note-simpkin-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-osipov_3-1" class="reference"><a href="#cite_note-osipov-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 7-5–7-8">: 7-5–7-8 </span></sup> Lanchester's laws have also been used to model <a href="/wiki/Guerrilla_warfare" title="Guerrilla warfare">guerrilla warfare</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The laws have also been applied to repeat battles with a range of inter-battle reinforcement strategies.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Attempts have been made to apply Lanchester's laws to conflicts between animal groups.<sup id="cite_ref-clifton20_15-0" class="reference"><a href="#cite_note-clifton20-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Examples include tests with chimpanzees<sup id="cite_ref-chimps_16-0" class="reference"><a href="#cite_note-chimps-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Ant" title="Ant">ants</a>. The chimpanzee application was relatively successful. A study of <a href="/wiki/Meat_ants" class="mw-redirect" title="Meat ants">Australian meat ants</a> and <a href="/wiki/Argentine_ant" title="Argentine ant">Argentine ants</a> confirmed the square law,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> but a study of <a href="/wiki/Fire_ants" class="mw-redirect" title="Fire ants">fire ants</a> did not confirm the square law.<sup id="cite_ref-plowes05_18-0" class="reference"><a href="#cite_note-plowes05-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Helmbold_Parameters">Helmbold Parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=7" title="Edit section: Helmbold Parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Helmbold Parameters offer precise numerical indices, grounded in historical data, for quickly and accurately comparing battles in terms of bitterness and the degree of advantage held by each side. While their definition is modeled after a solution of the Lanchester Square Law's differential equations, their numerical values are based entirely on the initial and final strengths of the opponents and in no way depend upon the validity of Lanchester's Square Law as a model of attrition during the course of a battle. </p><p>The solution of Lanchester's Square Law used here can be written as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a(t)&=\cosh(\lambda t)-\mu \sinh(\lambda t)\\d(t)&=\cosh(\lambda t)-\mu ^{-1}\sinh(\lambda t)\\\varepsilon &=\lambda T\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mi>T</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(t)&=\cosh(\lambda t)-\mu \sinh(\lambda t)\\d(t)&=\cosh(\lambda t)-\mu ^{-1}\sinh(\lambda t)\\\varepsilon &=\lambda T\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31eff7246500ebb2497ca20f84a702c7ad3f5b8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.251ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}a(t)&=\cosh(\lambda t)-\mu \sinh(\lambda t)\\d(t)&=\cosh(\lambda t)-\mu ^{-1}\sinh(\lambda t)\\\varepsilon &=\lambda T\end{aligned}}}" /></span> </p><p>Where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span> is the time since the battle began</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7931a26b0d360eaf90aa45247d2de5c984d5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.879ex; height:2.843ex;" alt="{\displaystyle a(t)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e115e742b9743138b3f4c1fd7cf9c7263dcdbe43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.865ex; height:2.843ex;" alt="{\displaystyle d(t)}" /></span> are the surviving fractions of the attacker's and defender's forces at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span> is the Helmbold intensity parameter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> is the Helmbold defender's advantage parameter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is the duration of the battle</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span> is the Helmbold bitterness parameter.</li></ul> <p>If the initial and final strengths of the two sides are known it is possible to solve for the parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf735e696f44dc83c9dea40fbc470ec2bd5fb4a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.675ex; height:2.843ex;" alt="{\displaystyle a(T)}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8591b695fb340bd2b3a711ea35fa6511f1aeec7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.661ex; height:2.843ex;" alt="{\displaystyle d(T)}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span>. If the battle duration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is also known, then it is possible to solve for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span>.<sup id="cite_ref-FOOTNOTEHelmbold1961a_19-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold1961a-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHelmbold1961b_20-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold1961b-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHelmbold2021app_A_21-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold2021app_A-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>If, as is normally the case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span> is small enough that the hyperbolic functions can, without any significant error, be replaced by their series expansion up to terms in the first power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span>, and if abbreviations adopted for the casualty fractions are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{A}=1-a(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}=1-a(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a600f6f21be247a5702b83d446b80cde7bf1e9bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.736ex; height:2.843ex;" alt="{\displaystyle F_{A}=1-a(T)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{D}=1-d(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{D}=1-d(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81fd943aa122bcae315e18cd70e8dfe351d66913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.85ex; height:2.843ex;" alt="{\displaystyle F_{D}=1-d(T)}" /></span>, then the approximate relations that hold include <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\sqrt {F_{A}F_{D}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\sqrt {F_{A}F_{D}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec721cf93914e0bd549671e91d8d74b9a2ba894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.553ex; height:3.509ex;" alt="{\displaystyle \varepsilon ={\sqrt {F_{A}F_{D}}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =F_{A}/F_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =F_{A}/F_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f549336bb3e7886b7b6e8ddcb57e68026fecdf0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.71ex; height:2.843ex;" alt="{\displaystyle \mu =F_{A}/F_{D}}" /></span>.<sup id="cite_ref-FOOTNOTEHelmbold202114–16,_app_A_22-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202114–16,_app_A-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> That <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span> is a kind of "average" (specifically, the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>) of the casualty fractions justifies using it as an index of the bitterness of the battle. </p><p>Statistical work prefers natural logarithms of the Helmbold Parameters. They are noted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b56fb083af41243dfe03ef5d6efb128bbaf37a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.761ex; height:2.676ex;" alt="{\displaystyle \log \mu }" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128540b80db5e4b291bb5e31fad0e5937b109966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.442ex; height:2.509ex;" alt="{\displaystyle \log \varepsilon }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3a5af333b359de02631e3a8b92e4c637236c16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.714ex; height:2.509ex;" alt="{\displaystyle \log \lambda }" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Major_findings">Major findings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=8" title="Edit section: Major findings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>See Helmbold (2021): </p> <ol><li>The Helmbold parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128540b80db5e4b291bb5e31fad0e5937b109966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.442ex; height:2.509ex;" alt="{\displaystyle \log \varepsilon }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b56fb083af41243dfe03ef5d6efb128bbaf37a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.761ex; height:2.676ex;" alt="{\displaystyle \log \mu }" /></span> are statistically independent, i.e., they measure distinct features of a battle.<sup id="cite_ref-FOOTNOTEHelmbold202118–19_23-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202118–19-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>The probability that the defender wins, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(D_{wins})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>i</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D_{wins})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a526fb4d50f39d17b8eba023c4fe95070b62b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.213ex; height:2.843ex;" alt="{\displaystyle P(D_{wins})}" /></span>, is related to the defender's advantage parameter via the <a href="/wiki/Logistic_function" title="Logistic function">logistic function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(D_{wins})=(1+\exp(-z))^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>i</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D_{wins})=(1+\exp(-z))^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a171775d5460c07770d3ed86cf75a057bd5f3dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.714ex; height:3.176ex;" alt="{\displaystyle P(D_{wins})=(1+\exp(-z))^{-1}}" /></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-0.1794+5.8694*\log \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>0.1794</mn> <mo>+</mo> <mn>5.8694</mn> <mo>∗</mo> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-0.1794+5.8694*\log \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ad9b75b5d4125f4602e3b4a2ad77a6ee0dbca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.709ex; height:2.676ex;" alt="{\displaystyle z=-0.1794+5.8694*\log \mu }" /></span>.<sup id="cite_ref-FOOTNOTEHelmbold202117–18_24-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202117–18-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> This logistic function is almost exactly skew-symmetric about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a11ea32b5bdb8358485ce79094eeea52d17b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.021ex; height:2.676ex;" alt="{\displaystyle \log \mu =0}" /></span>, rising from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(D_{wins})=0.1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>i</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D_{wins})=0.1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9fb9eec8c767605b00d1f17e85e765bd31b2d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle P(D_{wins})=0.1}" /></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu =-0.4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>0.4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu =-0.4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913be4aa081fbde9ac064d9d915b1ca990062eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.639ex; height:2.676ex;" alt="{\displaystyle \log \mu =-0.4}" /></span>, through <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(D_{wins})=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>i</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D_{wins})=0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3acd396be1018d74a6b9a01df767aaeedf9040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle P(D_{wins})=0.5}" /></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a11ea32b5bdb8358485ce79094eeea52d17b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.021ex; height:2.676ex;" alt="{\displaystyle \log \mu =0}" /></span>, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(D_{wins})=0.9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>i</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D_{wins})=0.9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267f25f79a87fa57915f11b0feb0c62401cb471b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle P(D_{wins})=0.9}" /></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \mu =+0.4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>μ</mi> <mo>=</mo> <mo>+</mo> <mn>0.4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \mu =+0.4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532697e6addf91ff6c59e721db3f045f0932c31c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.639ex; height:2.676ex;" alt="{\displaystyle \log \mu =+0.4}" /></span>. Because the probability of victory depends on the Helmbold advantage parameter rather than the force ratio, it is clear that force ratio is an inferior and untrustworthy predictor of victory in battle.</li> <li>While the defender's advantage varies widely from one battle to the next, on average it has been practically constant since 1600 CE.<sup id="cite_ref-FOOTNOTEHelmbold202120,_68–69_25-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202120,_68–69-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>Most of the other battle parameters (specifically the initial force strengths, initial force ratios, casualty numbers, casualty exchange ratios, battle durations, and distances advanced by the attacker) have changed so slowly since 1600 CE that only the most acute observers would be likely to notice any change over their nominal 50-year military career.<sup id="cite_ref-FOOTNOTEHelmbold202120,_app_C_26-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202120,_app_C-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>Bitterness (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128540b80db5e4b291bb5e31fad0e5937b109966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.442ex; height:2.509ex;" alt="{\displaystyle \log \varepsilon }" /></span>), casualty fractions (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dafcb50bfd139b3657b9ca92caa3cef2d195a163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.959ex; height:2.509ex;" alt="{\displaystyle F_{A}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0e98e9397b14b2f9399967a0c2154fa2c14d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.088ex; height:2.509ex;" alt="{\displaystyle F_{D}}" /></span> in the above notation), and intensity (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3a5af333b359de02631e3a8b92e4c637236c16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.714ex; height:2.509ex;" alt="{\displaystyle \log \lambda }" /></span>) also changed slowly before 1939 CE. But since then they have followed a startlingly steeper declining curve.<sup id="cite_ref-FOOTNOTEHelmbold202121,_app_C_part_4_27-0" class="reference"><a href="#cite_note-FOOTNOTEHelmbold202121,_app_C_part_4-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ol> <p>Some observers have noticed a similar post-WWII decline in casualties at the level of wars instead of battles.<sup id="cite_ref-lacina05_28-0" class="reference"><a href="#cite_note-lacina05-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lacina06_29-0" class="reference"><a href="#cite_note-lacina06-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lacina12a_30-0" class="reference"><a href="#cite_note-lacina12a-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lacina12b_31-0" class="reference"><a href="#cite_note-lacina12b-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Attrition_warfare" title="Attrition warfare">Attrition warfare</a></li> <li><a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a> similar mathematical model for predator-prey dynamics</li> <li><a href="/wiki/Maneuver_warfare" title="Maneuver warfare">Maneuver warfare</a></li> <li><a href="/wiki/Petrie_multiplier" title="Petrie multiplier">Petrie multiplier</a> similar mathematical model for sexism</li> <li><a href="/wiki/Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a></li> <li><a href="/wiki/Salvo_combat_model" title="Salvo combat model">Salvo combat model</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-lancs-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-lancs_1-0">^</a></b></span> <span class="reference-text">Lanchester F.W., <i>Mathematics in Warfare</i> in <i>The World of Mathematics,</i> Vol. 4 (1956) Ed. <a href="/wiki/James_R._Newman" title="James R. Newman">Newman, J.R.</a>, <a href="/wiki/Simon_%26_Schuster" title="Simon & Schuster">Simon and Schuster</a>, 2138–2157; anthologised from <i>Aircraft in Warfare</i> (1916)</span> </li> <li id="cite_note-davis95-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-davis95_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDavis1995" class="citation book cs1">Davis, Paul K. 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(1962). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/168104">"A Lanchester Model of Guerrilla Warfare"</a>. <i>Operations Research</i>. <b>10</b> (6): <span class="nowrap">818–</span>827. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.10.6.818">10.1287/opre.10.6.818</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0030-364X">0030-364X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/168104">168104</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Operations+Research&rft.atitle=A+Lanchester+Model+of+Guerrilla+Warfare&rft.volume=10&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E818-%3C%2Fspan%3E827&rft.date=1962&rft.issn=0030-364X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F168104%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1287%2Fopre.10.6.818&rft.aulast=Deitchman&rft.aufirst=S.+J.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F168104&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcCartney2022" class="citation journal cs1">McCartney, M (2022). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0378437121007500">"The solution of Lanchester's equations with inter-battle reinforcement strategies"</a>. <i>Physica A</i>. <b>586</b> (1): <span class="nowrap">1–</span>9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physa.2021.126477">10.1016/j.physa.2021.126477</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0378-4371">0378-4371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A&rft.atitle=The+solution+of+Lanchester%E2%80%99s+equations+with+inter-battle+reinforcement+strategies&rft.volume=586&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E9&rft.date=2022&rft_id=info%3Adoi%2F10.1016%2Fj.physa.2021.126477&rft.issn=0378-4371&rft.aulast=McCartney&rft.aufirst=M&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0378437121007500&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-clifton20-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-clifton20_15-0">^</a></b></span> <span class="reference-text">Clifton, E. (2020). A Brief Review on the Application of Lanchester's Models of Combat in Nonhuman Animals. Ecological Psychology, 32, 181-191. doi:10.1080/10407413.2020.1846456</span> </li> <li id="cite_note-chimps-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-chimps_16-0">^</a></b></span> <span class="reference-text">Wilson, M. L., Britton, N. F., & Franks, N. R. (2002). Chimpanzees and the mathematics of battle. Proceedings of the Royal Society B: Biological Sciences, 269, 1107-1112. doi:10.1098/rspb.2001.1926</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLymbery2023" class="citation journal cs1">Lymbery, Samuel J. (2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1073/pnas.2217973120">"Complex battlefields favor strong soldiers over large armies in social animal warfare"</a>. <i>PNAS</i>. <b>120</b> (37): e2217973120. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023PNAS..12017973L">2023PNAS..12017973L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.2217973120">10.1073/pnas.2217973120</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10500280">10500280</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37639613">37639613</a><span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PNAS&rft.atitle=Complex+battlefields+favor+strong+soldiers+over+large+armies+in+social+animal+warfare&rft.volume=120&rft.issue=37&rft.pages=e2217973120&rft.date=2023&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10500280%23id-name%3DPMC&rft_id=info%3Apmid%2F37639613&rft_id=info%3Adoi%2F10.1073%2Fpnas.2217973120&rft_id=info%3Abibcode%2F2023PNAS..12017973L&rft.aulast=Lymbery&rft.aufirst=Samuel+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1073%2Fpnas.2217973120&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-plowes05-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-plowes05_18-0">^</a></b></span> <span class="reference-text">Plowes, N. J. R., & Adams, E. S. (2005). An empirical test of Lanchester's square law: mortality during battles of the fire ant Solenopsis invicta. Proceedings of the Royal Society B: Biological Sciences, 272, 1809-1814. doi:10.1098/rspb.2005.3162</span> </li> <li id="cite_note-FOOTNOTEHelmbold1961a-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold1961a_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold1961a">Helmbold 1961a</a>.</span> </li> <li id="cite_note-FOOTNOTEHelmbold1961b-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold1961b_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold1961b">Helmbold 1961b</a>.</span> </li> <li id="cite_note-FOOTNOTEHelmbold2021app_A-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold2021app_A_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. app A.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202114–16,_app_A-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202114–16,_app_A_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 14–16, app A.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202118–19-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202118–19_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 18–19.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202117–18-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202117–18_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 17–18.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202120,_68–69-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202120,_68–69_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 20, 68–69.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202120,_app_C-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202120,_app_C_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 20, app C.</span> </li> <li id="cite_note-FOOTNOTEHelmbold202121,_app_C_part_4-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHelmbold202121,_app_C_part_4_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHelmbold2021">Helmbold 2021</a>, pp. 21, app C part 4.</span> </li> <li id="cite_note-lacina05-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-lacina05_28-0">^</a></b></span> <span class="reference-text">Lacina, Bethany & Nils Petter Gleditsch (2005) "Monitoring Trends in Flobal Combat: A New Dataset of Battle Deaths", Journal of Population (2005) 21:145-166</span> </li> <li id="cite_note-lacina06-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-lacina06_29-0">^</a></b></span> <span class="reference-text">Lacina, Bethany, Nils Petter Gleditsch, & Bruce Russett (2006) "The Declining Risk of Death in Battle", International Studies Quyarterly 50(3), 673-680</span> </li> <li id="cite_note-lacina12a-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-lacina12a_30-0">^</a></b></span> <span class="reference-text">Lacina, Bethany & Nils Petter Gleditsch, (2012) Journal of Conflict Resolution 57(6) 1109-1127</span> </li> <li id="cite_note-lacina12b-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-lacina12b_31-0">^</a></b></span> <span class="reference-text">Lacina, Bethany & Nils Petter Gleditsch, (2012) "The Waning of War Is Real: A Response to Gohdes and Price", Journal of Conflict Resolution</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=11" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Czarnecki, Joseph. <a rel="nofollow" class="external text" href="http://www.navweaps.com/index_tech/tech-076.htm">N-Squared Law: An Examination of one of the Mathematical Theories behind the Dreadnought Battleship</a> <i>Naval Weapons of the World</i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDupuy1979" class="citation book cs1">Dupuy, Col T N (1979). <i>Numbers, Predictions and War</i>. Macdonald and Jane's.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers%2C+Predictions+and+War&rft.pub=Macdonald+and+Jane%27s&rft.date=1979&rft.aulast=Dupuy&rft.aufirst=Col+T+N&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHelmbold1961a" class="citation book cs1">Helmbold, Robert L. (14 February 1961a). <i>Lanchester Parameters for Some Battles of the Last Two Hundred Years</i>. CORG Staff Paper CORG-SP-122.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lanchester+Parameters+for+Some+Battles+of+the+Last+Two+Hundred+Years&rft.pub=CORG+Staff+Paper+CORG-SP-122&rft.date=1961-02-14&rft.aulast=Helmbold&rft.aufirst=Robert+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHelmbold1961b" class="citation book cs1">Helmbold, Robert L. (1961b). "Lanchester's Equations, Historical Battles, and War Games". <i>Proceedings of the Eighth Military Operations Research Society Symposium, 18–20 October 1961</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lanchester%27s+Equations%2C+Historical+Battles%2C+and+War+Games&rft.btitle=Proceedings+of+the+Eighth+Military+Operations+Research+Society+Symposium%2C+18%E2%80%9320+October+1961&rft.date=1961&rft.aulast=Helmbold&rft.aufirst=Robert+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHelmbold2021" class="citation book cs1">Helmbold, Robert L. (12 May 2021). <i>The Key to Victory: Machine Learning the Lessons of History</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781668525289" title="Special:BookSources/9781668525289"><bdi>9781668525289</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Key+to+Victory%3A+Machine+Learning+the+Lessons+of+History&rft.date=2021-05-12&rft.isbn=9781668525289&rft.aulast=Helmbold&rft.aufirst=Robert+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLanchester1916" class="citation book cs1">Lanchester, Frederick W. (1916). <a class="external text" href="https://en.wikisource.org/wiki/Aircraft_in_Warfare_(1916)"><i>Aircraft in Warfare</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aircraft+in+Warfare&rft.date=1916&rft.aulast=Lanchester&rft.aufirst=Frederick+W.&rft_id=https%3A%2F%2Fen.wikisource.org%2Fwiki%2FAircraft_in_Warfare_%281916%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALanchester%27s+laws" class="Z3988"></span></li> <li>Niall J. MacKay <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.HO/0606300">Lanchester combat models</a>, <i>Mathematics Today</i>, 2006, Vol 42/5, pages 170–173.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lanchester%27s_laws&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.gamedeveloper.com/design/the-designer-s-notebook-kicking-butt-by-the-numbers-lanchester-s-laws">"Kicking Butt By the Numbers: Lanchester's Laws"</a>, a Designer's Notebook column by Ernest Adams in the <a rel="nofollow" class="external text" href="http://www.gamedeveloper.com">Game Developer</a> website</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Lanchester%27s_laws&oldid=1261805434">https://en.wikipedia.org/w/index.php?title=Lanchester%27s_laws&oldid=1261805434</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Differential_equations" title="Category:Differential equations">Differential equations</a></li><li><a href="/wiki/Category:Predation" title="Category:Predation">Predation</a></li><li><a href="/wiki/Category:Combat_modeling" title="Category:Combat modeling">Combat modeling</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_uses_Russian-language_script_(ru)" title="Category:CS1 uses Russian-language script (ru)">CS1 uses Russian-language script (ru)</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_the_United_States_Army" title="Category:Wikipedia articles incorporating text from the United States Army">Wikipedia articles incorporating text from the United States Army</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_public_domain_works_of_the_United_States_Government" title="Category:Wikipedia articles incorporating text from public domain works of the United States Government">Wikipedia articles incorporating text from public domain works of the United States Government</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_July_2016" title="Category:Use dmy dates from July 2016">Use dmy dates from July 2016</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 December 2024, at 01:26<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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