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


"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, JulesAntoine 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 lifelong 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

http://wwwhistory.mcs.stand.ac.uk/history/Curves/Lissajous.html

http://mathworld.wolfram.com/LissajousCurve.html

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.


