Editing Tautochrone curve (section)

== The tautochrone problem ==
[[File:Huygens - Horologium oscillatorium, sive De motu pendulorum ad horologia aptato demonstrationes geometricae, 1673 - 869780.jpeg|thumb|upright|[[Christiaan Huygens]], ''[[Horologium Oscillatorium|Horologium oscillatorium sive de motu pendulorum]]'', 1673]]
{{Quote box|width=30%|
quote=It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.
|source=''[[Moby Dick]]'' by [[Herman Melville]], 1851}}
The tautochrone problem, the attempt to identify this curve, was solved by [[Christiaan Huygens]] in 1659. He proved geometrically in his ''[[Horologium Oscillatorium]]'', originally published in 1673, that the curve is a [[cycloid]].

{{Blockquote|On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other&nbsp;...<ref>{{cite book |last=Blackwell |first=Richard J. |title=Christiaan Huygens' The Pendulum Clock |publisher=Iowa State University Press |date=1986 |location=Ames, Iowa |isbn=0-8138-0933-9 |at= Part II, Proposition XXV, p.&nbsp;69}}</ref>}}

The cycloid is given by a point on a circle of radius <math>r</math> tracing a curve as the circle rolls along the <math>x</math> axis, as:
<math display="block">\begin{align}
  x &= r(\theta - \sin \theta) \\
  y &= r(1 - \cos \theta),
 \end{align}</math>

Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as diameter of the circle that generates the cycloid, multiplied by <math>\pi / 2</math>. In modern terms, this means that the time of descent is <math display="inline">\pi \sqrt{r/g}</math>, where <math>r</math> is the radius of the circle which generates the cycloid, and <math>g</math> is the [[gravity of Earth]], or more accurately, the earth's gravitational acceleration.

[[File:Isochronous cycloidal pendula.gif|thumb|Five isochronous cycloidal pendulums with different amplitudes]]
This solution was later used to solve the problem of the [[brachistochrone curve]].  [[Johann Bernoulli]] solved the problem in a paper (''[[Acta Eruditorum]]'', 1697).

[[File:CyloidPendulum.png|right|thumb|Schematic of a [[cycloidal pendulum]]]]
The tautochrone problem was studied by Huygens more closely when it was realized that a pendulum, which follows a circular path, was not [[isochronous]] and thus his [[pendulum clock]] would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved unhelpful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock [[escapement]]s could greatly reduce this source of inaccuracy.

Later, the mathematicians [[Joseph Louis Lagrange]] and [[Leonhard Euler]] provided an analytical solution to the problem.