Editing Pendulum (section)

=== 1673: Huygens' ''Horologium Oscillatorium'' ===
In 1673, 17 years after he invented the pendulum clock, [[Christiaan Huygens]] published his theory of the pendulum, ''[[Horologium Oscillatorium|Horologium Oscillatorium sive de motu pendulorum]]''.<ref>{{cite web
  | last = Huygens
  | first = Christian
  |author2=translated by Ian Bruce
  | title = Horologium Oscillatorium
  | website = 17centurymaths
  | publisher = 17thcenturymaths.com
  | date = July 2007
  | url = http://www.17centurymaths.com/contents/huygenscontents.html
  | format = PDF
  | access-date = 2009-03-01}}</ref><ref>The constellation of [[Horologium (constellation)|Horologium]] was later named in honor of this book.</ref>  [[Marin Mersenne]] and [[René Descartes]] had discovered around 1636 that the pendulum was not quite isochronous; its period increased somewhat with its amplitude.<ref name="Matthews">{{cite book
  | last = Matthews
  | first = Michael R.
  | title = Science Teaching: The Role of History and Philosophy of Science
  | publisher = Psychology Press
  | year = 1994
  | pages = 121–122
  | url = https://books.google.com/books?id=qnwzRqh5jFMC&q=mersenne+isochronous+pendulum&pg=PA121
  | isbn = 978-0-415-90899-3}}</ref>  Huygens analyzed this problem by determining what curve an object must follow to descend by gravity to the same point in the same time interval, regardless of starting point; the so-called ''[[tautochrone problem|tautochrone curve]]''. By a complicated method that was an early use of [[calculus]], he showed this curve was a [[cycloid]], rather than the circular arc of a pendulum,<ref>[http://www.17centurymaths.com/contents/huygens/horologiumpart2b.pdf Huygens, Horologium Oscillatorium], Part 2, Proposition 25</ref> confirming that the pendulum was not isochronous and Galileo's observation of isochronism was accurate only for small swings.<ref>{{cite web
 |last=Mahoney 
 |first=Michael S. 
 |date=March 19, 2007 
 |url=http://www.princeton.edu/~mike/articles/huygens/timelong/timelong.html 
 |title=Christian Huygens: The Measurement of Time and of Longitude at Sea 
 |publisher=Princeton University 
 |access-date=2007-05-27 
 |archive-url=https://web.archive.org/web/20071204152637/http://www.princeton.edu/~mike/articles/huygens/timelong/timelong.html 
 |archive-date=December 4, 2007 
 |url-status=dead 
}}</ref>  Huygens also solved the problem of how to calculate the period of an arbitrarily shaped pendulum (called a ''compound pendulum''), discovering the ''[[Center of percussion|center of oscillation]]'', and its interchangeability with the pivot point.<ref>{{cite conference
  | first = Fabio
  | last = Bevilaqua
  |author2=Lidia Falomo |author3=Lucio Fregonese |author4=Enrico Gianetto |author5=Franco Giudise |author6=Paolo Mascheretti
  | title = The pendulum: From constrained fall to the concept of potential
  | book-title = The Pendulum: Scientific, Historical, Philosophical, and Educational Perspectives
  | pages = 195–200
  | publisher = Springer
  | year = 2005
  | url = https://books.google.com/books?id=3GV2NgDwtjMC&pg=PA195
  | isbn = 1-4020-3525-X
  | access-date = 2008-02-26}} gives a detailed description of Huygens' methods</ref>

The existing clock movement, the [[verge escapement]], made pendulums swing in very wide arcs of about 100°.<ref name="Headrick">{{cite journal
 |last=Headrick 
 |first=Michael 
 |year=2002 
 |title=Origin and Evolution of the Anchor Clock Escapement 
 |periodical=Control Systems Magazine, Inst. Of Electrical and Electronic Engineers 
 |volume=22 
 |issue=2 
 |url=http://www.geocities.com/mvhw/anchor.html 
 |access-date=2007-06-06 
 |archive-url=https://web.archive.org/web/20091025120920/http://geocities.com/mvhw/anchor.html
 |archive-date=October 25, 2009
 |url-status=dead
}}</ref> Huygens showed this was a source of inaccuracy, causing the period to vary with amplitude changes caused by small unavoidable variations in the clock's drive force.<ref>"''...it is affected by either the intemperance of the air or any faults in the mechanism so the crutch QR is not always activated by the same force... With large arcs the swings take longer, in the way I have explained, therefore some inequalities in the motion of the timepiece exist from this cause...''",  {{cite book
  | last =Huygens
  | first = Christiaan
  | title = Horologium
  | publisher = Adrian Vlaqc
  | year = 1658
  | location = The Hague
  | url = http://www.antique-horology.org/_Editorial/Horologium/Horologium.pdf
  }}, translation by Ernest L. Edwardes (December 1970) ''Antiquarian Horology'', Vol.7, No.1</ref>  To make its period isochronous, Huygens mounted cycloidal-shaped metal guides next to the pivots in his clocks, that constrained the suspension cord and forced the pendulum to follow a cycloid arc (see [[cycloidal pendulum]]).<ref name="Andrewes1994">Andrewes, W.J.H. [https://books.google.com/books?id=F7wNQk219KMC&pg=PA126 ''Clocks and Watches: The leap to precision''] in {{cite book
  | first = Samuel
  | last = Macey
  | title = Encyclopedia of Time
  | pages = 123–125
  | publisher = Taylor & Francis
  | year = 1994
  | isbn = 978-0-8153-0615-3}}</ref>  This solution didn't prove as practical as simply limiting the pendulum's swing to small angles of a few degrees. The realization that only small swings were [[isochronous]]  motivated the development of the [[anchor escapement]] around 1670, which reduced the pendulum swing in clocks to 4°–6°.<ref name="Headrick" /><ref>[https://books.google.com/books?id=xuDDqqa8FlwC&pg=PA312 Usher, 1988], p.312</ref> This became the standard escapement used in pendulum clocks.