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Lissajous or Bowditch Curves

Parametric Equations

Note:  The  n  determines the number of complete loops.

The Slide Show on the right illustrates different values for the constant terms.

Replay the animation


Also, these curves are easily entered and modified on a graphing calculator.
          American Nathaniel Bowditch

"I never come across one of Laplace's 'Thus it plainly appears' without feeling sure that I have hours of hard work before me to fill up the chasm and find out and show how it plainly appears."
Nathaniel Bowditch  

Historical Sketch

Nathaniel Bowditch (1773 - 1838) was the first American to receive international recognition as a mathematician.  Moreover, he was the first to investigate a family of curves now usually named for the French physicist, Jules-Antoine Lissajous.  Lissajous independently published his work much later in 1857.

Bowditch, working in the isolation of New England's Salem and Boston areas, held a life-long fascination with doing tedious calculations.  He learned Latin and several other languages in order to read the mathematical publications being imported from Europe.  In particular, he is known to have studied Newton's Principia and Laplace's Mécanique celeste, an important guidebook for astronomers. 

Be reminded that a copy of Euclid's Elements was not printed in the newly independent country until 1803 in Philadephia.  During Bowditch’s lifetime in Boston, one printed mathematics book would commonly be imported by a professor.  Students were then expected to learn from notes and notebooks hand copied in the classroom.   This atmosphere may have contributed to Bowditch's refusal of chairs of mathematics at several universities.  He preferred being president of the Essex Fire and Marine Insurance Company (1804 - 1823).

Also be reminded that the principal means of international communication as well as commerce was the open sea.  Mathematics was a navigational necessity.  In 1802 Bowditch published the New American Practical Navigator.  This book became known as "the seaman's Bible" and is still published.

Lissajous (1822 - 1880) is thought to have encountered these curves when he collected data for his optical method of studying vibrations.

Useful Links and Books
Boyer, Carl B., revised by U. C. Merzbach, A History of Mathematics, 2nd ed., John Wiley and Sons, 1991.
Eves, Howard, An Introduction to the History of Mathematics, 6th ed,. The Saunders College Publishing, 1990.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998.
Sobel, Dava,  Longitude:  The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, Walker and Co., 1998.
Yates, Robert,  CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952.
MATHEMATICA® Code and animation contributed by
Gus Gordillo, 2004.