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Brachistochrone #77
Brachistochrone #58
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Deposit #88

Al Lehnen
Madison Area Technical College
Madison, WI

Brachistochrone model
The Spighi Model
Istituto e Museo di Storia della Scienza, in Florence, Italy

i.e., (Johann, John, or Jean)

Joh. Bernoulli

and his illustrations of the

and the "Brachysochrona". . .


from the famous article published in
the 1697 "Acta Eruditorium."


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Brachistochrone ~ Cycloid
A Comparison of Times of Fastest Descent

This section . . . .

Brachistochrone-cycloid animation

The Brachistochrone-Cycloid Revisited:  A Timely Consideration

Brachistochrone Equations

( Please open the "pdf" file for the continuing text and graphs.)

Figure 7
Graph of descent times

For Lehnan's description of Deposit # 88 . . .
MAA button
Lehnan presented his work
at MathFest 2008.

Carr, George S., Formulas and Theorems in PURE MATHEMATICS, 2nd ed., Chelsea Publishing Co., New York, 1970. Reprint of Carr's 1886 edition under the title of A Synopsis of Elementary Results in Pure Mathematics, London and Cambridge.
N. P. Johnson,"The Brachistochrone Problem", The College Mathematics Journal 35, 192-197, 2004.
W. S. Kimball, Calculus of Variations, Butterworths Scientific Publications, London, 1952, p. 328.
Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998, p. 1027.
Lockwood, E. H., A Book of Plane Curves, Campridge University Press, 1961, pp. 167-171.
Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 589 - 590.  See Evolute.
J. Zeng, "A Note on the Brachistochrone Problem", The College Mathematics Journal 27, 206-208, 1996.

The National Curve Bank thanks Al Lehnen for Deposit #88.  < >
Lehnen signature
  The NCB also thanks the Istituto e Museo di Storia della Scienza of  Florence, Italy and the Huntington Library of San Marino, California for permission to use the illustrations.
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