Editing Inclined plane (section)

==History==

{| class="toccolours" style="float: margin-left: 1em; margin-right: 1em; font-size: 100%; background:#c6dbf7; color:black; width: 40%; float: right;" cellspacing="3"
| style="text-align: center;" |'''Stevin's proof'''
|-
| style="text-align: left;" |[[Image:StevinEquilibrium.svg|center|150px]] In 1586, Flemish engineer [[Simon Stevin]] (Stevinus) derived the mechanical advantage of the inclined plane by an argument that used a string of beads.<ref name="Koetsier">{{cite conference
  | first = Teun
  | last = Koetsier
  | title = Simon Stevin and the rise of Archimedean mechanics in the Renaissance
  | book-title = The Genius of Archimedes – 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference Held at Syracuse, Italy, June 8–10, 2010
  | pages = 94–99
  | publisher = Springer
  | year = 2010
  | url = https://books.google.com/books?id=65Pz4_XJrgwC&pg=PA95
  | isbn = 978-90-481-9090-4
  }}</ref> He imagined two inclined planes of equal height but different slopes, placed back-to-back as in a prism (''A, B, C'' above).  A loop of string with beads at equal intervals is draped over the inclined planes, with part of the string hanging down below. The beads resting on the planes act as loads on the planes, held up by the tension force in the string at point ''T''. Stevin's argument goes like this:<ref name="Koetsier" /><ref name="Devreese">{{cite book   
  | last = Devreese
  | first = Jozef T. 
  |author2=Guido Vanden Berghe
  | title = 'Magic is no magic': The wonderful world of Simon Stevin
  | publisher = WIT Press
  | year = 2008
  | pages = 136–139
  | url = https://books.google.com/books?id=f59h2ooQGmcC&pg=PA136
  | isbn = 978-1-84564-391-1}}</ref><ref name="Feynman">{{cite book   
  | last = Feynman
  | first = Richard P. 
  |author2=Robert B. Leighton |author3=Matthew Sands
   | title = The Feynman Lectures on Physics, Vol. I
  | publisher = California Inst. of Technology
  | year = 1963
  | location = USA
  | pages = 4.4–4.5
  | url = https://www.feynmanlectures.caltech.edu/I_04.html#Ch4-S2-p9
  | isbn = 978-0-465-02493-3}}</ref>
*The string must be stationary, in [[static equilibrium]]. If the string was heavier on one side than the other, and began to slide right or left under its own weight, when each bead had moved to the position of the previous bead the string would be indistinguishable from its initial position and therefore would continue to be unbalanced and slide.  This argument could be repeated indefinitely, resulting in a circular [[perpetual motion]], which is absurd. Therefore, it is stationary, with the forces on the two sides at point ''T'' (''above'') equal.
*The portion of the chain hanging below the inclined planes is symmetrical, with an equal number of beads on each side. It exerts an equal force on each side of the string.  Therefore, this portion of the string can be cut off at the edges of the planes ''(points S and V)'', leaving only the beads resting on the inclined planes, and this remaining portion will still be in static equilibrium.
*Since the beads are at equal intervals on the string, the total number of beads supported by each plane, the total load, is proportional to the length of the plane.  Since the input supporting force, the tension in the string, is the same for both, the mechanical advantage of each plane is proportional to its slant length
As pointed out by Dijksterhuis,<ref>E.J.Dijksterhuis: ''Simon Stevin'' 1943</ref> Stevin's argument is not completely tight. The forces exerted by the hanging part of the chain need not be symmetrical because the hanging part ''need not retain its shape'' when let go. Even if the chain is released with a zero angular momentum, motion including oscillations is possible unless the chain is initially in its equilibrium configuration, a supposition which would make the argument circular. 
|}

Inclined planes have been used by people since prehistoric times to move heavy objects.<ref name="Conn">Therese McGuire, ''Light on Sacred Stones'', in {{cite book   
  | last = Conn
  | first = Marie A. 
  |author2=Therese Benedict McGuire
  | title = Not etched in stone: essays on ritual memory, soul, and society
  | publisher = University Press of America
  | year = 2007
  | pages = 23
  | url = https://books.google.com/books?id=kEPkDyvek3sC&pg=PA23
  | isbn = 978-0-7618-3702-2}}</ref><ref name="Dutch">{{cite web
  | last = Dutch
  | first = Steven
  | title = Pre-Greek Accomplishments
  | work = Legacy of the Ancient World
  | publisher = Prof. Steve Dutch's page, Univ. of Wisconsin at Green Bay
  | year = 1999
  | url = http://www.uwgb.edu/dutchs/westtech/xancient.htm
  | access-date = March 13, 2012
  | archive-date = August 21, 2016
  | archive-url = https://web.archive.org/web/20160821064729/http://www.uwgb.edu/dutchs/westtech/xancient.htm
  | url-status = dead
  }}</ref> The sloping roads and [[causeway]]s built by ancient civilizations such as the Romans are examples of early inclined planes that have survived, and show that they understood the value of this device for moving things uphill. The heavy stones used in ancient stone structures such as [[Stonehenge]]<ref name="Moffett">{{cite book   
  | last = Moffett
  | first = Marian 
  |author2=Michael W. Fazio |author3=Lawrence Wodehouse
   | title = A world history of architecture
  | publisher = Laurence King Publishing
  | year = 2003
  | pages = 9
  | url = https://books.google.com/books?id=IFMohetegAcC&pg=PT8
  | isbn = 978-1-85669-371-4}}</ref> are believed to have been moved and set in place using inclined planes made of earth,<ref name="Peet">{{cite book   
  | last = Peet
  | first = T. Eric 
  | title = Rough Stone Monuments and Their Builders
  | publisher = Echo Library
  | year = 2006
  | pages = 11–12
  | url = https://books.google.com/books?id=2c15PS0uwHEC&q=slope
  | isbn = 978-1-4068-2203-8}}</ref> although it is hard to find evidence of such temporary building ramps.  The [[Egyptian pyramids]] were constructed using inclined planes,<ref name="Thomas">{{cite web
  | last = Thomas
  | first = Burke
  | title = Transport and the Inclined Plane
  | work = Construction of the Giza Pyramids
  | publisher = world-mysteries.com
  | year = 2005
  | url = http://www.world-mysteries.com/gw_tb_gp.htm
  | access-date = March 10, 2012
  | archive-date = March 13, 2012
  | archive-url = https://web.archive.org/web/20120313035958/http://www.world-mysteries.com/gw_tb_gp.htm
  | url-status = dead
  }}</ref><ref name="Isler">{{cite book   
  | last = Isler
  | first = Martin 
  | title = Sticks, stones, and shadows: building the Egyptian pyramids
  | publisher = University of Oklahoma Press
  | year = 2001
  | location = USA
  | pages = [https://archive.org/details/sticksstonesshad00mart/page/211 211]–216
  | url = https://archive.org/details/sticksstonesshad00mart
  | url-access = registration
  | isbn = 978-0-8061-3342-3}}</ref><ref name="SpragueDeCamp">{{cite book   
  | last =  Sprague de Camp
  | first = L.
  | title = The Ancient Engineers
  | publisher = Barnes & Noble
  | year = 1990
  | location = USA
  | pages = 43
  | url = https://books.google.com/books?id=cauMt9vJLs0C&q=ramp
  | isbn = 978-0-88029-456-0}}</ref> [[Siege]] ramps enabled ancient armies to surmount fortress walls.   The ancient Greeks constructed a paved ramp 6&nbsp;km (3.7 miles) long, the [[Diolkos]], to drag ships overland across the [[Isthmus of Corinth]].<ref name="Silverman" />

However the inclined plane was the last of the six classic [[simple machine]]s to be recognised as a machine. This is probably because it is a passive and motionless device (the load is the moving part),<ref name="Reuleaux" /> and also because it is found in nature in the form of slopes and hills. Although they understood its use in lifting heavy objects, the [[Ancient Greece|ancient Greek]] philosophers who defined the other five simple machines did not include the inclined plane as a machine.<ref>for example, the lists of simple machines left by Roman architect [[Marcus Vitruvius Pollio|Vitruvius]] (c. 80 – 15 BCE) and Greek philosopher [[Heron of Alexandria]] (c. 10 – 70 CE) consist of the five classical simple machines, excluding the inclined plane. –   {{cite book   
  | last = Smith
  | first = William 
  | title = Dictionary of Greek and Roman antiquities
  | publisher = Walton and Maberly; John Murray
  | year = 1848
  | location = London
  | pages = 722
  | url = https://books.google.com/books?id=zfIrAAAAYAAJ&q=%22inclined+plane%22+%22mechanical+powers%22+greek&pg=PA722
  }},  {{cite book 
|last=Usher 
|first=Abbott Payson  
|title=A History of Mechanical Inventions 
|publisher=Courier Dover Publications 
|year=1988 
|location=USA 
|pages=98, 120
|url=https://books.google.com/books?id=xuDDqqa8FlwC&q=wedge+and+screw&pg=PA196 
|isbn=978-0-486-25593-4
}}</ref> This view persisted among a few later scientists; as late as 1826 [[Karl Christian von Langsdorf|Karl von Langsdorf]] wrote that an inclined plane "''...is no more a machine than is the slope of a mountain''".<ref name="Reuleaux">Karl von Langsdorf (1826) ''Machinenkunde'', quoted in  {{cite book   
  | last = Reuleaux
  | first = Franz 
  | title = The kinematics of machinery: Outlines of a theory of machines
  | publisher = MacMillan
  | year = 1876
  | pages = [https://archive.org/details/kinematicsmachi01reulgoog/page/n524 604]
  | url = https://archive.org/details/kinematicsmachi01reulgoog
  }}</ref> The problem of calculating the force required to push a weight up an inclined plane (its mechanical advantage) was attempted by Greek philosophers [[Heron of Alexandria]] (c. 10 - 60 CE) and [[Pappus of Alexandria]] (c. 290 - 350 CE), but their solutions were incorrect.<ref>{{cite book   
  | last =  Heath
  | first = Thomas Little
  | title = A History of Greek Mathematics, Vol. 2
  | publisher = The Clarendon Press
  | year = 1921
  | location = UK
  | pages = [https://archive.org/details/bub_gb_7DDQAAAAMAAJ/page/n365 349], 433–434
  | url = https://archive.org/details/bub_gb_7DDQAAAAMAAJ
  }}</ref><ref name="Laird">Egidio Festa and [[Sophie Roux]], ''The enigma of the inclined plane'' in {{cite book   
  | last = Laird
  | first = Walter Roy 
  |author2=Sophie Roux|author2-link=Sophie Roux
  | title = Mechanics and natural philosophy before the scientific revolution
  | publisher = Springer
  | year = 2008
  | location = USA
  | pages = 195–221
  | url = https://books.google.com/books?id=z3pRa83qz2IC&q=stevin+&pg=PA209
  | isbn = 978-1-4020-5966-7}}</ref><ref name="Meli">{{cite book   
  | last = Meli
  | first = Domenico Bertoloni 
  | title = Thinking With Objects: The Transformation of Mechanics in the Seventeenth Century
  | publisher = JHU Press
  | year = 2006
  | pages = 35–39
  | url = https://books.google.com/books?id=I6QreZN02joC&q=inclined+plane
  | isbn = 978-0-8018-8426-9}}</ref>

It was not until the [[Renaissance]] that the inclined plane was solved mathematically and classed with the other simple machines. The first correct analysis of the inclined plane appeared in the work of 13th century author [[Jordanus de Nemore]],<ref name="Boyer">{{cite book   
  | last = Boyer
  | first = Carl B. 
  |author2=Uta C. Merzbach|author2-link= Uta Merzbach
  | title = A History of Mathematics, 3rd Ed.
  | publisher = John Wiley and Sons
  | year = 2010
  | url = https://books.google.com/books?id=BokVHiuIk9UC&q=%22inclined+plane%22+stevin+jordanus+galileo&pg=PT243
  | isbn = 978-0-470-63056-3}}</ref><ref name="Usher">{{cite book   
  | last = Usher
  | first = Abbott Payson 
  | title = A History of Mechanical Inventions
  | publisher = Courier Dover Publications
  | year = 1988
  | pages = 106
  | url = https://books.google.com/books?id=xuDDqqa8FlwC&q=inclined+plane&pg=PA106
  | isbn = 978-0-486-25593-4}}</ref> however his solution was apparently not communicated to other philosophers of the time.<ref name="Laird" /> [[Girolamo Cardano]] (1570) proposed the incorrect solution that the input force is proportional to the angle of the plane.<ref name="Koetsier" /> Then at the end of the 16th century, three correct solutions were published within ten years, by Michael Varro (1584), [[Simon Stevin]] (1586), and [[Galileo Galilei]] (1592).<ref name="Laird" /> Although it was not the first, the derivation of Flemish engineer [[Simon Stevin]]<ref name="Meli" /> is the most well-known, because of its originality and use of a string of beads (see box).<ref name="Feynman" /><ref name="Boyer" /> In 1600, Italian scientist Galileo Galilei included the inclined plane in his analysis of simple machines in ''Le Meccaniche'' ("On Mechanics"), showing its underlying similarity to the other machines as a force amplifier.<ref name="Machamer">{{cite book   
  | last = Machamer
  | first = Peter K. 
  | title = The Cambridge Companion to Galileo
  | publisher = Cambridge University Press
  | year = 1998
  | location = London
  | pages = 47–48
  | url = https://books.google.com/books?id=1wEFPLoqTeAC&q=%22inclined+plane%22+galileo+Meccaniche&pg=PA48
  | isbn = 978-0-521-58841-6}}</ref>

The first elementary rules of sliding [[friction]] on an inclined plane were discovered by [[Leonardo da Vinci]] (1452-1519), but remained unpublished in his notebooks.<ref name="Armstrong">{{cite book   
  | last = Armstrong-Hélouvry
  | first = Brian 
  | title = Control of machines with friction
  | publisher = Springer
  | year = 1991
  | location = USA
  | pages = 10
  | url = https://books.google.com/books?id=0zk_zI3xACgC&q=friction+leonardo+da+vinci+amontons+coulomb&pg=PA10
  | isbn = 978-0-7923-9133-3}}</ref> They were rediscovered by [[Guillaume Amontons]] (1699) and were further developed by [[Charles-Augustin de Coulomb]] (1785).<ref name="Armstrong" /> [[Leonhard Euler]] (1750) showed that the [[Tangent (trigonometric function)|tangent]] of the [[angle of repose]] on an inclined plane is equal to the [[coefficient of friction]].<ref name="Meyer">{{cite book   
  | last = Meyer
  | first = Ernst 
  | title = Nanoscience: friction and rheology on the nanometer scale
  | publisher = World Scientific
  | year = 2002
  | pages = 7
  | url = https://books.google.com/books?id=Rhi7odTe2BEC&q=%22Leonhard+euler%22+angle+%22inclined+plane%22&pg=PA7
  | isbn = 978-981-238-062-3}}</ref>