|The NCB invites
you to join Galileo, Newton, Leibniz, Huygens,
and two Bernoullis in thinking about one of the most celebrated
of 17th century mathematics. The
"brachistochrone" is from the Greek meaning "shortest" and
What is the path
- curve - producing the shortest possible
time for a particle to
descend from a given point to another point not
directly below the start? Will it be a
straight line, an arc of a
circle, or just what? Will it be a minimum of a function?
The shortest distance between two points is
a line, but the descent of a weighted particle is acted upon, in the
very least, by gravity. A bead descending a wire is often
used to depict the pathway, but investigators usually ignore
The correct answer
is that a body takes less time to fall along the arc of a circumference
than to fall along the "line" of a corresponding chord.
The cycloid path
allows the particle to move rapidly at first, while
in steep descent, and thus build up sufficient speed to overcome the
greater distance the particle must travel. Thus, the speed of the
descending particle is
accelerated by gravity.
We begin by introducing a MATHEMATICA® animation.
animations feature several
the corresponding brachistochrone.
You also need
to be familiar with the cycloid. A cycloid is
the locus of a point on the circumference of a circle rotating along a
fixed line . . . .
Law for the
is often applied in
deriving the equations.
|A lengthy discussion of the equations
is found on two other pages.
A reproduction of the original Acta figures (1697) is found on Brachistochrone
Other MATHEMATICA® Animations
Useful Links and Books
brachistochrone and cycloid have a very rich math and physics
The National Curve Bank also has MAPLE
animations of the cycloid family of curves.
John (Jean, Johann)
A large statue of Leibniz is at
the Royal Academy of Arts
in the heart of fashionable London.
Note the English chose to spell
his name as . . . .
|Beckmann, Petr, A History of π (PI), St.
Martin's Press, 1971, pp. 139-140.
|Burton, David M., The History of Mathematics, 5th
ed., McGraw Hill, 2003, p. 446.
brachistochrone is for "the shrewdest
mathematicians of all the world."
Bernoulli, June, 1696
| Eves, Howard, AN
INTRODUCTION TO THE HISTORY OF MATHEMATICS, 6th ed.,
Saunders College Publishing, 1992, p. 426.
|Johnson, Nils P., The
Brachistochrone Problem, The College
Mathematics Journal, vol. 35 (3), May 2004, pp.
|A cycloid has the
noteworthy property that the time it takes from any point along it to
its lowest point does not depend on the starting point.
Johnson and others
suggest the Euler-Lagrange formula and boundary conditions applied to
the brachistochrone will define a differential equation whose solution
is similar to finding critical points (usually a maximum or minimum) in
ordinary calculus. "Thus a global problem - an integral over a
path - has a local constraint - a differential equation -which dictates
the solution at each point along the way."
|Lockwood, E. H., A Book of CURVES, Cambridge
University Press, 1961, p. 88.
|Katz, Victor J., A
History of Mathematics, 2nd ed., Addison Wesley Longman, 1998,
pp. 547-549, 562.
Katz, Victor J., A History of
Mathematics, Brief ed., Pearson Addison Wesley, 2004, pp. 250,
|Simmons, George F., Calculus Gems: Brief Lives and Memorable
Mathematics, McGraw-Hill,1992, pp. 308-313.
vivid, enthusiastic, personal style is in sharp contrast to the dead,
gray, impersonal style of most of the writing in scientific journals
we admire Huygens because he first discovered that a heavy particle
slides down to the bottom of a cycloid in the same time, no matter
where it starts. But you will be petrified with astonishment when
say that this very same cycloid, the tautochorone of Huygens is also
the brachistochrone we are seeking."
|Szapiro, Ben, Revisiting A Classic Least Time Problem,
|Stewart, James, Calculus, 5th ed.,
2003, p. 691.
|"One of the first people to study the
cycloid was Galileo, who proposed that bridges be built in the shape of
cycloids and who tried to find the area under one arch of a cycloid."
|Wagon, Stan, MATHEMATICA®IN
ACTION, W. H. Freeman and Co., 1991, pp. 60-66 and 385-389.
0-7167-2229-1 or ISBN 0-7167-2202-X
Wagon, Stan, MATHEMATICA® IN ACTION, 2nd ed.,
Springer-Verlag, 2000. ISBN 0-387-98684-7 for other animations.
|Weisstein, Eric W., CRC Concise Encyclopedia of MATHEMATICS,
CRC Press, 1999.
was one of the earliest problems posed in the Calculus of Variations."
|Yates, Robert C., Curves and Their Properties,
NCTM, 1952, pp. 68-69.
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