Editing Logistic function (section)

== History ==
[[File:Courbe_logistique,_Verhulst,_1845.png|thumb|upright|300px|Original image of a logistic curve, contrasted with what Verhulst called a "logarithmic curve" (in modern terms, "exponential curve")]]

The logistic function was introduced in a series of three papers by [[Pierre François Verhulst]] between 1838 and 1847, who devised it as a model of [[population growth]] by adjusting the [[exponential growth]] model, under the guidance of [[Adolphe Quetelet]].{{sfn|Cramer|2002|pp=3–5}} Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838,<ref name=verhulst1838 /> then presented an expanded analysis and named the function in 1844 (published 1845);{{efn|1=The paper was presented in 1844, and published in 1845: "(Lu à la séance du 30 novembre 1844)." "(Read at the session of 30 November 1844).", p. 1.}}<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1845| title = Recherches mathématiques sur la loi d'accroissement de la population | journal = Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles |volume = 18 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323640_0018&DMDID=dmdlog7| access-date = 18 February 2013|trans-title= Mathematical Researches into the Law of Population Growth Increase |page=[https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify={%22pages%22:%5B21%5D,%22view%22:%22info%22} 8] |quote=Nous donnerons le nom de ''logistique'' à la courbe [We will give the name ''logistic'' to the curve]}}</ref> the third paper adjusted the correction term in his model of Belgian population growth.<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1847| title = Deuxième mémoire sur la loi d'accroissement de la population | journal = Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique |volume = 20| pages = 1–32 |doi=10.3406/marb.1847.3457 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323659_0020&DMDID=dmdlog29| access-date = 18 February 2013}}</ref>

The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth approaches the limit with an exponentially decaying gap, like the initial stage in reverse.

Verhulst did not explain the choice of the term "logistic" ({{langx|fr|link=no|logistique}}), but it is presumably in contrast to the ''logarithmic'' curve,<ref>{{cite journal |title=Math-alive! using original sources to teach mathematics in social context |journal=[[PRIMUS (journal)|PRIMUS]] |volume=8 |first=Bonnie |last=Shulman |pages=1–14 |issue=March |year=1998 |doi=10.1080/10511979808965879 |url=https://www.researchgate.net/publication/233238354 |quote=The diagram clinched it for me: there two curves labeled "Logistique" and "Logarithmique" are drawn on the same axes, and one can see that there is a region where they match almost exactly, and then diverge.<br/>I concluded that Verhulst's intention in naming the curve was indeed to suggest this comparison, and that "logistic" was meant to convey the curve's "log-like" quality.}}</ref>{{efn|1=Verhulst first refers to arithmetic ''progression'' and geometric ''progression'', and refers to the geometric growth curve as a ''logarithmic'' curve (confusingly, the modern term is instead ''exponential'' curve, which is the inverse). He then calls his curve ''logistic'', in contrast to ''logarithmic'', and compares the logarithmic curve and logistic curve in the figure of his paper.}} and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of [[arithmetic growth]] and [[geometric growth]] (whose curve he calls a [[logarithmic curve]], instead of the modern term [[exponential curve]]), and thus "logistic growth" is presumably named by analogy, ''logistic'' being from {{langx|grc|λογιστικός|logistikós}}, a traditional division of [[Greek mathematics]].{{efn|1=In Ancient Greece, {{lang|grc|λογιστικός}} referred to practical computation and accounting, in contrast to {{lang|grc|ἀριθμητική}} (''{{lang|grc-Latn|arithmētikḗ}}''), the theoretical or philosophical study of numbers. Confusingly, in English, ''[[arithmetic]]'' refers to practical computation, even though it derives from {{lang|grc|ἀριθμητική}}, not {{lang|grc|λογιστικός}}. See for example [[Louis Charles Karpinski]], ''Nicomachus of Gerasa: Introduction to Arithmetic'' (1926) p. 3: "Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after [[Pythagoras]], however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, ''λογιστική'' [accounting or practical logistic] ... In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children."}} 

As a word derived from ancient Greek mathematical terms,<ref name=tts>{{cite journal|first1= J.|last1=Tepic|first2= I.|last2=Tanackov|first3=Gordan|last3= Stojić |title = Ancient logistics – historical timeline and etymology|journal = Technical Gazette |volume =18|year =2011 |url= http://pdfs.semanticscholar.org/fb9f/f927aa0472a4df2635b8b221c4c67d567046.pdf|archive-url= https://web.archive.org/web/20190309153711/http://pdfs.semanticscholar.org/fb9f/f927aa0472a4df2635b8b221c4c67d567046.pdf|url-status= dead|archive-date= 2019-03-09|issue =3|s2cid=42097070}}</ref>
the name of this function is unrelated to the military and management term ''logistics'', which is instead from {{langx|fr|{{wikt-lang|fr|logis}}}} "lodgings",<ref>{{cite book |title=Tableau Analytique des principales combinaisons De La Guerre, Et De Leurs Rapports Avec La Politique Des États: Pour Servir D'Introduction Au Traité Des Grandes Opérations Militaires |author=Baron de Jomini |author-link=Antoine-Henri Jomini |year=1830 |page= [https://books.google.com/books?id=ofAcWX9UsIUC&q=logistique&pg=PA74 74]}}</ref> though some believe the Greek term also influenced ''logistics'';<ref name=tts/> see {{slink|Logistics|Origin}} for details.